Estimation and Hypothesis Testing of Fixed Effects Models-Based Uncertainty for Factor Designs

Estimation and Hyp othesis T esting of Fixed Eects Mo dels-Based Uncertain t y for F actor Designs F an Zhang, Zhiming Li ∗ Col le ge of Mathematics and System Scienc e, Xinjiang University, Urumqi 830046, China Abstract: T o analyze the uncertain data frequently encoun tered in practice, this pap er prop oses nov el xed-eects mo dels that incorp orate an uncertain measure to inv estigate v ariables of interest and n uisance v ariables in factor designs. First, an uncertain xed- eects (UFE) mo del of a single-factor design is established, and uncertain estimation and h yp othesis testing are conducted. W e then extend the UFE mo del to t wo-factor designs with and without in teractions and classify them as balanced or un balanced based on the equalit y of replicates within each com bination. In the ab o v e UFE mo dels, the eectiveness and practicalit y of estimation and hypothesis metho ds are demonstrated through three real- w orld cases, including b oth balanced and unbalanced designs. These examples highlight the mo dels’ abilit y to handle uncertain exp erimen tal data. Keyw ords: F actor design, xed eects, uncertain measure, uncertain mo del. 1 In tro duction The design of experiments is widely applied in agriculture (T orres-Mercado et al. (2024)), medicine (Muguruma et al. (2022)), engineering (Liew et al. (2022)), manufacturing (Rah- matabadi et al. (2023)), and scien tic researc h (Alexanderian (2021)). An experiment often in volv es altering one or more factors that ma y aect the outcome under con trolled con- ditions, and these designs are referred to as single-factor or multi-factor designs. Its core lies in systematically arranging exp erimen ts to inv estigate the relationships b et ween the researc h sub ject and inuencing factors, thereby providing a reliable foundation for data analysis. T o analyze signican t dierences among factor eects, xed-eects, random- eects, or mixed-eects mo dels are commonly used in multi-factor designs. The xed- eects (FE) mo del is a panel-data regression mo del. Unlik e the FE mo del, the random eects mo del treats individual-sp ecic heterogeneity as part of the error terms, pioneered b y Eisenhart (1947). The mixed-eects mo del com bines the rst t wo t yp es (Laird and W are (1982)). Among these mo dels, xed eects are regarded as a ubiquitous and p o w erful to ol for eliminating un wan ted v ariation (Breuer and Dehaa (2024)). ∗ Corresp onding author: zmli@xju.edu.cn 1 F ollo wing Dean et al. (2017), we review the classical FE mo del within the framework of probabilit y theory . T ak e a single-factor design as an example. Let y ij b e the j th ( j = 1 , . . . , n ) observ ation taken under the i th ( i = 1 , . . . , a ) level or treatment of a factor A . Let µ b e the ov erall mean of all treatmen ts, and µ i ( i = 1 , . . . , a ) b e the mean of the i th level of the factor A . Under the assumptions of indep endent and iden tically distributed observ ations, and normalit y , no corrections, and homoskedasticit y of random errors, y ij = µ i + ε ij ( i = 1 , . . . , a, j = 1 , . . . , n ) is called a mean mo del, where ε ij is a random error and ε ij ∼ N (0 , σ 2 ) . Denote a i = µ − µ i ( i = 1 , . . . , a ) , a parameter unique to the i th level called the i th level eect. Th us, an alternative mo del, called the FE mo del, is established b y y ij = µ + a i + ε ij , i = 1 , . . . , a, j = 1 , . . . , n, a  i =1 a i = 0 , whic h is also called the one-w ay or single-factor analysis of v ariance mo del. This frame- w ork w as formalized by Sc heé (1959) and extended to multi-factor designs, in which analysis of v ariance (ANOV A) tests the signicance of factor eects. Subsequen t research has addressed key complexities: handling unbalanced data via metho ds such as exact p er- m utation tests and generalized linear mo dels (Kherad-Pajouh and Renaud (2010); Thiel et al. (2017)); addressing high-dimensional problems via regularization techniques (Engel et al. (2020)); managing heteroscedasticity with robust methods like the v ariance-w eighted F -appro ximation and its renements (W elc h (1951); Alexander and Gov ern (1994); Pilk- ington et al. (2024)); emplo ying developed m ultiple comparison pro cedures (T uk ey (1949); Hsu (1996); Mee and Hui (2023)); and utilizing residual analysis techniques for mo del diagnostics (Anscombe and T ukey (1963); Mon tgomery (2017)). Numerous theoretical ac hievemen ts and applications of FE mo dels can b e referenced in Wilk and Kempthorne (1955), Pietraszek et al. (2016), and P asserine and Breitkreitz (2024). Ho w ev er, some c hallenges arise in exp erimental observ ations: small sample sizes by design and frequent inheren t uncertaint y , rather than genuine randomness, render probabilit y-based statisti- cal to ols less applicable. Empirical evidence suggests that real-world data often fail to satisfy core assumptions in probability theory (including normality , frequency stability , in- dep endence, and randomness). When the sample size is limited, the data are imprecise, or uncertain ty originates from human b elief and cognition, probability theory may no longer b e adequate in real-world problems, as noted in numerous studies, such as carb on sp ot price (Liu and Li (2024)), O 3 concen tration (Xia and Li (2025)), epidemic spreading (Xie and Lio (2024)), and p opulation (Y ang and Liu (2024)). The existence of these problems necessitates no vel measures to provide a more suitable mo deling framew ork. T o solve the c hallenges men tioned ab ov e, Liu (2007) rst prop osed an uncertain measure grounded in the axioms of normality , dualit y , subadditivity , and pro duct, further establishing uncertain t y theory . As a notable practical extension of this theory , Liu (2010) pioneered the use of uncertain statistics to address the collection, analysis, and in terpretation of uncertain data in real-world applications. Curren tly , uncertain regression mo dels are p opular in the eld of uncertain statistics. Uncertain statistical inference for these mo dels encompasses t wo core tasks: parameter estimation and hypothesis testing. F or uncertain regression mo dels, the estimation metho ds include least squares (Y ao and Liu 2 (2018)), momen t (Lio and Liu (2018)), uncertain maximum lik eliho o d (Lio and Liu (2020)), least absolute deviations estimation (Liu and Y ang (2019)), and ridge estimation (Zhang and Gao (2024)). F or some complex regression mo dels, lo cal p olynomial and B-spline estimation, as w ell as tw o-step estimation, are employ ed (Ding and Zhang (2021), Zhang and Li (2025a)). On the other hand, hypothesis testing under uncertaint y is emplo yed to v alidate the results. Unlike probabilistic metho ds that rely on con v entional statistics, this approach directly utilizes ra w data. Y e and Liu (2022) in tro duced the framework for testing regression co ecients in uncertain mo dels, demonstrating its sup eriorit y in handling non-probabilistic residuals. F urther adv ances by Y e and Liu (2023) extended the metho d to multiv ariate settings, demonstrating robust p erformance when sto c hastic tests fail. Moreo ver, (Zhang and Li, 2025b) conducted uncertaint y-based homogeneit y and common tests for multiple nite p opulations, thereby broadening the applicability of inference under uncertain ty . Based on the abov e analysis, n umerous v aluable researc h con tributions ha ve b een made to the eld of uncertain regression mo dels. Uncertain regression mo dels based on exp er- imen tal design often present a distinct yet closely related scenario—it inv olv es dierent treatmen t com binations, balanced/unbalanced structures, and presp ecied p otential in- teraction eects. Nevertheless, the underlying uncertaint y in such designs has received little atten tion, and existing uncertain regression to olb o xes do not accommo date these fea- tures. T o bridge this gap, this pap er systematically introduces uncertaint y theory into the FE framework of exp erimen tal design. T o v alidate eectiveness, we reanalyse “presumed random” data with uncertain mo dels. The main innov ations are summarised as follows: (i) A system of uncertain xed eects (UFE) mo dels is rst prop osed for b oth single- and tw o-factor designs, thereby providing a dedicated theoretical framework for analyzing uncertain exp erimen tal data. (ii) Parameter estimation metho ds, in tegrating generalized linear mo dels with uncertain v ectors, are prop osed for unbalanced designs, which ov ercame the limitation of traditional least squares and ac hieved explicit estimation and derived distributions. (iii) The homogeneity test of eects in UFE mo dels is conducted by adjusting data for xed eects to main tain inter-group indep endence, thereb y ensuring reliable results. (iv) Through three diverse real examples spanning dieren t designs, w e substantiate the suitabilit y and practical utility of the prop osed metho dologies. As summarized in T able 1, the UFE mo del adv ances beyond traditional probabilistic FE mo dels by fundamen tally not relying on frequency stability or randomness assumptions. This shift confers sev eral key adv an tages: (a) it oers enhanced tolerance to extreme v alues through its hea vy-tailed normal uncertaint y distribution (Guo et al. (2010)), making it b etter suited for small-sample scenarios; (b) it employs a stricter rejection criterion based on direct ev aluation of raw data against the null hypothesis distribution, providing higher sensitivit y to detect systematic dierences often missed b y traditional metho ds; and (c) its applicability in scenarios with sub jectiv e uncertain t y or unv eriable assumptions complemen ts traditional approaches, particularly when data are limited. The remainder of this pap er is organized as follows. Section 2 establishes an uncer- tain xed-eects mo del for a single-factor design, inv estigates parameter estimation, and explores the testing pro cess for related hypotheses. The parameter estimation and uncer- 3 T able 1: Comparison of probabilistic and uncertain xed eects (FE) mo dels. Asp ect Probabilistic FE Mo del Ucertain FE Mo del Measure Probabilit y measure Uncertain measure Error Normal distribution (thin-tailed) Normal uncertain t y distribution (hea vy-tailed) Hyp otheses T est statistics ( F /t test) Ra w data, in verse uncertaint y distribution Sensitivit y in small samples Lo w (low statistical p o w er) High (stricter rejection criterion) Applicable scenarios Large samples, ideal assumptions satised Small samples, violated assumptions, sub jectiv e uncertain ty tain h yp otheses across tw o factors are analyzed in Section 3, accounting for interaction eect and replication v ariabilit y . Section 4 demonstrates the practical eectiveness of the prop osed metho ds through three real examples, and Section 5 provides a brief conclusion. 2 Uncertain xed-eects mo del with single factor In this section, w e rst review the basic concept of an uncertain measure (Liu (2023)), and then prop ose a xed-eects mo del with uncertaint y for single-factor designs. 2.1 Basic concepts Let Γ is a nonempty set, and L b e a σ -algebra o ver Γ . A map M : L 7→ [0 , 1] is called an uncertain measure, satisfying four axioms: (i) (Normalit y axiom) M { Γ } = 1 for the universal set Γ . (ii) (Dualit y axiom) M { Λ } + M { Λ c } = 1 for an y ev ent Λ . (iii) (Subadditivit y axiom) F or every countable sequence of even ts Λ 1 , Λ 2 ,..., w e hav e M  ∞  i =1 Λ i  ≤ ∞  i =1 M { Λ i } . (iv) (Pro duct axiom) Let ( Γ i , L i , M i ) b e a sequence of uncertaint y spaces and Λ i are arbitrarily chosen even ts from L i for i = 1, 2,..., then the pro duct uncertain measure M is an uncertain measure satisfying M { ∞  i =1 Λ i } = ∞  i =1 M { Λ i } . F or any Borel set B ⊆ R , ξ is called an uncertain v ariable if the set { γ ∈ Γ | ξ ( γ ) ∈ B } ∈ L . The function Φ( z ; θ ) = M { ξ ≤ z } ( z ∈ R ) is called an uncertain ty distribution with paramater θ ( ∈ Θ) of ξ . Under regular conditions, the inv erse function of Φ( z ; θ ) is called an inv erse uncertaint y distribution, denoted by Φ − 1 ( α ; θ ) for α ∈ (0 , 1) . An uncertain v ariable ξ is normal if its uncertain t y distribution and inv erse function are dened as: 4 Φ( z ; µ, σ ) =  1 + exp  π ( µ − z ) √ 3 σ  − 1 , z ∈ R , Φ − 1 ( α ; µ, σ ) = µ + √ 3 σ π ln  α 1 − α  , α ∈ (0 , 1) , where e and σ are real num b ers with σ > 0 , denoted b y ξ ∼ N ( e, σ ) . When e = 0 and σ = 1 , ξ is a standard normal uncertaint y distribution. A regular uncertain t y distribution family { Φ( z ; θ ) : θ ∈ Θ } is said to b e nonem b edded for θ 0 ∈ Θ at lev el α if Φ − 1 ( β ; θ 0 ) > Φ − 1 ( β ; θ ) or Φ − 1 (1 − β ; θ ) > Φ − 1 (1 − β ; θ 0 ) , some θ ∈ Θ , 0 < β ≤ α/ 2 . The normal uncertaint y distribution family { N ( e, σ ) : e ∈ R , σ > 0 } is nonembedded for an y ( e 0 , σ 0 ) at a signicance lev el α (Y e and Liu (2022)). 2.2 Uncertain xed-eects mo del In an exp eriment, supp ose there is only one factor A with r lev els A 1 , A 2 , . . . , A r . Let m i b e the rep eated times at lev el A i for i = 1 , 2 , . . . , r , with a total of N =  r i =1 m i . If m 1 = m 2 = · · · = m r = m , it is called a balanced design. Otherwise, it is unbalanced. Let z ij ( i = 1 , . . . , r , j = 1 , . . . , m i ) b e observ ations of the j th replicate at the i th level. Based on this, an uncertain mo del of a single-factor design is dened as follows z ij = µ i + ε ij , i = 1 , 2 , . . . , r ; j = 1 , 2 , . . . , m i , (1) where µ i is the mean of A i , and ε ij are mutually indep endent and identically distributed uncertain errors for the j th exp eriment at the i th level. Theorem 1. In the mo del (1), let ˆ µ i ( i = 1 , ..., r ) b e the le ast-squar es estimator of µ i . Then, ˆ µ i = 1 m i  m i j =1 z ij . Pr o of. By emplo ying the least-squares metho d (Y ao and Liu (2018)), ˆ µ i ( i = 1 , ..., r ) is obtained b y solving ˆ µ = argmin µ r  i =1 m i  j =1 ( z ij − µ i ) 2 , where ˆ µ = ( ˆ µ 1 , ˆ µ 2 , ..., ˆ µ r ) . T o p erform subsequent condence interv al construction and hypothesis testing, w e as- sume ε ij ∼ N (0 , σ 0 ) , where σ 0 is determined and v alidated through the residual analysis pro cedure describ ed in Remark 1. Remark 1. After obtaining the p oint estimates, we verify the distribution of al l r esiduals ˆ ε ij = z ij − ˆ µ i in mo del (1) to justify ε ij ∼ N (0 , σ 0 ) . First, it is c onrme d that ˆ ε ij ∼ N (0 , σ i 0 ) using Cor ol lary 1 in Y e and Liu (2022). Then, fol lowing Zhang and Li (2025b), two tests ar e c onducte d on the standar d deviation: (i) an unc ertain homo geneity test σ 1 = σ 2 = · · · = σ r = σ ; (ii) an unc ertain c ommon test σ = σ 0 under (i) holds. Only after these validations c an unc ertain c ondenc e intervals and hyp othesis tests b e p erforme d. The same pr o c e dur e applies to the subse quent mo dels and thus is not r ep e ate d her e after. 5 Since ε ij ∼ N (0 , σ 0 ) , it is obvious that z ij ∼ N ( µ i , σ 0 ) . By the op erational la w, ˆ µ i has the uncertain ty distribution N ( µ i , σ 0 ) , i.e., Φ( z ; µ i , σ 0 ) =  1 + exp  π ( µ i − z ) √ 3 σ 0  − 1 . Then, follo wing Lio and Liu (2018), the α (e.g., 95%) condence interv al of µ i is C I ( µ i ) =  Φ − 1  1 − α 2 ; ˆ µ i , σ 0  , Φ − 1  1 + α 2 ; ˆ µ i , σ 0  = ˆ µ i ± σ 0 √ 3 π ln  1 + α 1 − α  . (2) Let w i b e the w eigh t of the total n umber of exp erimen ts contributed b y the replications of lev el A i , giv en by w i = m i / N ( i = 1 , 2 , . . . , r ) . Denote µ = r  i =1 w i µ i , r  i =1 w i = 1 . W e call a i = µ i − µ ( i = 1 , 2 , . . . , r ) the eect of level A i , which is the deviation of the mean of lev el A i from the ov erall mean. Obviously ,  r i =1 w i a i = 0 . Thus, the uncertain mo del (1) is equiv alen t to the mo del      z ij = µ + a i + ε ij , i = 1 , 2 , . . . , r , j = 1 , 2 , . . . , m i , r  i =1 w i a i = 0 , ε ij ∼ N (0 , σ 0 ) are m utually indep enden t . (3) W e refer to the mo del (3) as an uncertain xed-eects (UFE) model based on a single-factor design, denoted b y the SUFE mo del. Theorem 2. In the SUFE mo del (3), let ˆ µ and ˆ a i ( i = 1 , . . . , r ) b e the le ast-squar es estimators of µ and a i . Then, ˆ µ = 1 N r  i =1 m i  j =1 z ij , ˆ a i =  1 m i − 1 N  m i  j =1 z ij − 1 N  k  = i m k  j =1 z kj . (4) Pr o of. By employing the least-squares metho d, ˆ µ and ˆ a i ( i = 1 , . . . , r ) are obtained by solving ( ˆ µ, ˆ a ) = argmin µ, a r  i =1 m i  j =1 ( z ij − µ − a i ) 2 , where ˆ a = (ˆ a 1 , ˆ a 2 , ..., ˆ a r ) . W e can obtain ˆ µ = 1 N r  i =1 m i  j =1 z ij , ˆ a i = 1 m i m i  j =1 z ij − 1 N r  i =1 m i  j =1 z ij . T o facilitate the direct deriv ation of their uncertain t y distributions and to pro vide a basis for constructing uncertain condence interv als, we express each estimator as a sum of indep enden t uncertain normal v ariables, which yields (4). 6 By the op erational law, it follows that ˆ µ ∼ N ( µ, σ 0 ) and ˆ a i ∼ N ( a i , 2 (1 − m i / N ) σ 0 ) , i.e., Φ( z ; µ, σ 0 ) =  1 + exp  π ( µ − z ) √ 3 σ 0  − 1 , Φ  z ; a i , 2  1 − m i N  σ 0  =  1 + exp  π ( a i − z ) 2 √ 3  1 − m i N  σ 0  − 1 . Then, the α condence interv als of µ and a i are C I ( µ ) =  Φ − 1  1 − α 2 ; ˆ µ, σ 0  , Φ − 1  1 + α 2 ; ˆ µ, σ 0  = ˆ µ ± σ 0 √ 3 π ln  1 + α 1 − α  , (5) C I ( a i ) =  Φ − 1  1 − α 2 ; ˆ a i , σ a i  , Φ − 1  1 + α 2 ; ˆ a i , σ a i  = ˆ a i ± σ a i √ 3 π ln  1 + α 1 − α  , (6) where σ a i = 2 (1 − m i / N ) σ 0 . 2.3 Uncertain h yp otheses In this subsection, we in v estigate the equalit y of the means in the uncertain mo del of a single-factor design and equalit y of the eects in uncertain xed-eects models. In this and subsequen t sections concerning h yp othesis testing, for simplicity , only unknown parameters are retained in the inv erse distribution according to tw o cases: (i) if the mean e 0 is known, Φ − 1 ( α ; e 0 , σ j 0 ) is replaced by Φ − 1 ( α ; σ j 0 ) , and (ii) if the standard deviation σ 0 is known, Φ − 1 ( α ; a j 0 , σ 0 ) is rewritten b y Φ − 1 ( α ; a j 0 ) . F or the mo del (1), the homogeneity h yp otheses of means µ i : H a : µ 1 = µ 2 = · · · = µ r v ersus H b : µ 1 , µ 2 , . . . , µ r are not all equal . If the n ull h yp othesis H a is rejected at the signicance level α , the lev els of factor A dier signican tly . Otherwise, there is no signican t dierence. Since a i = µ i − µ ( i = 1 , 2 , . . . , r ) , the h yp otheses H a is equiv alen t to the following case H 0 : a 1 = a 2 = · · · = a r v ersus H 1 : a 1 , . . . , a r are not all equal . (7) Under the constraint  r i =1 w i a i = 0 , if the null hypothesis holds, i.e., a 1 = a 2 = · · · = a r ≜ a , then the constraint b ecomes a  r i =1 w i = a · 1 = 0 , which implies a = 0 . Under the constrain t, the h yp othesis a 1 = a 2 = · · · = a r is equiv alten t to a 1 = a 2 = · · · = a r = 0 . Consequen tly , the homogeneit y test of the eects a i is sucien t to determine whether all eects are zero. The pro cess can b e extended to b oth balanced and unbalanced designs. Let z ij ( i = 1 , . . . , r ; j = 1 , . . . , m i ) b e observ ations satisfying z ij ∼ N ( µ i , σ i ) . Since indep enden t samples of ˆ a i are not directly observ able, we rely on the original data z ij to construct the rejection region. A natural adjustment based on a i = µ i − µ is ˜ z ij = z ij − ˆ µ, where ˆ µ ∼ N ( µ, σ 0 ) . How ev er, the estimator ˆ µ is dep endent on groups, b ecause ˜ z ij and ˜ z kl ( k 6 = i ) b ecome correlated through ˆ µ . 7 T o preserve indep endence, z ij should b e adjusted only by a xed constant. Denote a ∗ i = a i + ( µ − µ 0 ) . W e therefore replace ˆ µ with a xed constant µ 0 that can b e estimated from the data but is treated as known in the testing pro cedure, yielding ˜ z ij = z ij − µ 0 = a i + µ − µ 0 + ε ij = a ∗ i + ε ij . Then, ˜ z ij are m utually indep enden t and follo w N ( a ∗ i , σ i ) . Theorem 3. F or the hyp othesis pr oblem (7), the r eje ction r e gion at signic anc e level α is W =  ( ˜ z i 1 , ˜ z i 2 , . . . , ˜ z im i ) : ∃ i 6 = j ( i, j ∈ { 1 , 2 , ..., r } ) such that at le ast α of indexes p ’s with 1 ≤ p ≤ m i satisfy ˜ z ip < Φ − 1 ( α/ 2; a j 0 ) or ˜ z ip > Φ − 1 (1 − α/ 2; a j 0 )  , wher e Φ − 1 ( α ; a j 0 ) = a j 0 + σ i 0 √ 3 π ln  α 1 − α  and a i 0 and σ i 0 ar e estimators of a ∗ i , σ i . Pr o of. Since µ 0 is a xed constant in the testing pro cedure, the adjustments ˜ z ij = a ∗ i + ε ij , are indep endent and ˜ z ij ∼ N ( a ∗ i , σ i ) . Th us, the hypothesis ( 7) is equiv alent to a ∗ 1 = a ∗ 2 = · · · = a ∗ r . Note that the normal uncertaint y distribution family { N ( e, σ ) : e ∈ R } is nonem b edded for all parameters at an y signicance lev el α . By Theorem 4 of Zhang and Li (2025b), w e obtain the rejection region W at signicance lev el α . Remark 2. In the mo dels (1) and (3), we use a single σ 0 for estimating various ee cts. It aims to simplify c omputation and aligns with the SUFE mo del’s assumption of homo gene ous standar d deviations, ther eby inte gr ating information fr om al l exp erimental units to yield mor e stable estimates. However, in the homo geneity test of me ans or ee cts, we use individual σ i for e ach level to avoid the risk that a single σ 0 might obscur e true ee cts, ther eby impr oving the pr e cision of the tests and the r eliability of the test r esults. Remark 3. F or al l homo geneity tests, the r eje ction criterion of the nul l hyp othesis H 0 c an b e e quivalently expr esse d via the ac c eptanc e interval (AI). Sp e cic al ly, for a dataset ( z i 1 , z i 2 , . . . , z im i ) , the ac c eptanc e interval for a p ar ameter θ j 0 is dene d as: AI ( z i ; θ j 0 ) =  Φ − 1 ( α/ 2; θ j 0 ) , Φ − 1 (1 − α/ 2; θ j 0 )  . If any z i has mor e than αm i data outside its c orr esp onding AI ( z i ; θ j 0 ) , H 0 is r eje cte d. 3 Uncertain xed-eects mo del with t w o factors Let A and B b e tw o factors of interest in an exp eriment, eac h ha ving A 1 , A 2 , . . . , A r and B 1 , B 2 , . . . , B s , where r and s are the num b er of levels. Eac h com bination pairs one lev el of A with one level of B , written as A i B j ( i = 1 , 2 , . . . , r , j = 1 , 2 , . . . , s ). The exp eriment is rep eated m ij times for each combination A i B j , with a total of N =  r i =1  s j =1 m ij . Let z ij l denote observ ations of the l th ( l = 1 , 2 , . . . , m ij ) unit of A i B j , and all z ij l from distinct A i B j are indep endent. W e consider t w o cases in the following subsections: (i) tw o-factor design without in teraction, and (ii ) t wo-factor design with interaction. 8 3.1 T w o-factor design without in teraction In the tw o-factor mo del, assume the t w o factors are indep enden t and that their eects are additive, with no interactions. Let µ i · ( i = 1 , 2 , . . . , r ) and µ · j ( j = 1 , 2 , . . . , s ) b e the mean of A i and B j , and µ b e the o v erall mean. Th us, the mean of a combination A i B j is expressed as: µ ij = µ + a i + b j , i = 1 , 2 , . . . , r , j = 1 , 2 , . . . , s, where a i = µ i · − µ , and b j = µ · j − µ are the main eects of factors A and B . Denote m i · =  s j =1 m ij and m · j =  r i =1 m ij . Let w i · and w · j b e the prop ortion of total observ ations that b elong to A i and B j ; that is, w i · = m i ./ N , w · j = m · j / N . In the case of a balanced design, w i · = 1 /r , w · j = 1 /s . Then, a UFE mo del without interaction is dened as follows:      z ij l = µ + a i + b j + ε ij l , i = 1 , 2 , . . . , r , j = 1 , 2 , . . . , s, l = 1 , 2 , . . . , m ij , r  i =1 w i · a i = 0 , s  j =1 w · j b j = 0 , ε ij l ∼ N (0 , σ 0 ) are m utually indep enden t . (8) Theorem 4. In the mo del (8) with a b alanc e d design, i.e., m ij = m ( i = 1 , . . . , r, j = 1 , . . . , s ) , let ˆ µ , ˆ a i , and ˆ b j b e the le ast-squar es estimators of µ , a i , and b j . Then, ˆ µ = 1 r sm r  i =1 s  j =1 m  l =1 z ij l , ˆ a i = r − 1 r sm s  j =1 m  l =1 z ij l − 1 r sm  k  = i s  j =1 m  l =1 z kj l , (9) ˆ b j = s − 1 r sm r  i =1 m  l =1 z ij l − 1 r sm r  i =1  k  = j m  l =1 z ikl . (10) Pr o of. By employing the least-squares metho d, ˆ µ , ˆ a i , and ˆ b j ( i = 1 , . . . , r, j = 1 , . . . , s ) are obtained b y solving ( ˆ µ, ˆ a , ˆ b ) = argmin µ, a , b r  i =1 s  j =1 m  l =1 ( z ij l − µ − a i − b j ) 2 , where ˆ a = (ˆ a 1 , ˆ a 2 , . . . , ˆ a r ) and ˆ b = ( ˆ b 1 , ˆ b 2 , . . . , ˆ b s ) . Hence, ˆ a i = 1 sm s  j =1 m  l =1 z ij l − 1 r sm r  i =1 s  j =1 m  l =1 z ij l , ˆ b j = 1 r m r  i =1 m  l =1 z ij l − 1 r sm r  i =1 s  j =1 m  l =1 z ij l . (11) F urther, w e write eac h estimator as a sum of indep endent normal uncertain v ariables, so that the op erational law can b e applied to derive their uncertaint y distributions. It follows from the op erational law that ˆ µ ∼ N ( µ, σ 0 ) , ˆ a i ∼ N ( a i , 2(1 − 1 /r ) σ 0 ) , and ˆ b j ∼ N ( b j , 2(1 − 1 /s ) σ 0 ) . Then, the α condence interv als of µ, a i and b j are C I ( µ ) = ˆ µ ± σ 0 √ 3 π ln  1 + α 1 − α  , (12) 9 C I ( a i ) = ˆ a i ± σ a i √ 3 π ln  1 + α 1 − α  , C I ( b j ) = ˆ b j ± σ b j √ 3 π ln  1 + α 1 − α  , (13) where σ a i = 2(1 − 1 /r ) σ 0 and σ b j = 2(1 − 1 /s ) σ 0 . Theorem 4 is only a v ailable to the balanced case, but not to the unbalanced one. Therefore, in the case of imbalance, the Lagrangian multiplier metho d is prop osed to estimate the parameters in an un balanced mo del (8). Let Z b e an N × 1 resp onse v ector, X b e an N × p matrix for p = r + s + 1 , that is, Z = ( z 111 , . . . , z 11 m 11 , . . . , z 1 s 1 , . . . , z 1 sm 1 s , . . . , z r 11 , . . . , z r 1 m r 1 , . . . , z rs 1 , . . . , z rsm rs ) T , X = ( x 111 , . . . , x 11 m 11 , . . . , x 1 s 1 , . . . , x 1 sm 1 s , . . . , x r 11 , . . . , x r 1 m r 1 , . . . , x rs 1 , . . . , x rsm rs ) T suc h that x ij l β = µ + a i + b j , where x ij l = (1 , 0 , . . . , 1 , . . . , 0 , 0 , . . . , 1 , . . . , 0) 1 × p , corre- sp onding to parameters β = ( µ, a 1 , . . . , a r , b 1 , . . . , b s ) T . Denote ε = ( ε 1 , . . . , ε N ) T . Th us, the mo del (8) is equiv alen t to the following system Z = X β + ε , C β = d , (14) where C =  0 w 1 · · · · w 2 · 0 · · · 0 0 0 · · · 0 w · 1 · · · w · 2  2 × p , d = (0 , 0) T . Theorem 5. In the mo del (14) with an unb alanc e d design, let ˆ β b e the le ast-squar es estimators of the p ar ameter ve ctor β . Then, ˆ β = ( X T X ) + X T Z − 1 2 ( X T X ) + C T ˆ λ , (15) wher e ˆ λ is the ve ctor of estimate d L agr ange multipliers, and A + denotes the gener alize d inverse of matrix A . Pr o of. By emplo ying the Lagrangian m ultiplier metho d, ˆ β and ˆ λ are obtained b y solving ( ˆ β , ˆ λ ) = argmin β , λ [( Z − X β ) T ( Z − X β ) + λ T ( C β − d )] . Through calculation, we hav e − 2 X T ( Z − X ˆ β ) + C T ˆ λ = 0 , C ˆ β = d , which yields the result. T o obtain condence interv als of the parameters, we need their distributions of the least-squares estimators. Theorem 6. In the mo del (14) with an unbalanced design, the following results hold. (i) ε ∼ N (0 , σ 0 I N ) , where I N is the N × N iden tity matrix. (ii) Z ∼ N ( X β , σ 0 I N ) . (iii) ˆ µ ∼ N ( µ,  N j =1 | q 1 j | σ 0 ) , ˆ a i ∼ N ( a i ,  N j =1 | q i +1 ,j | σ 0 ) , ˆ b j ∼ N ( b j ,  N k =1 | q r + j +1 ,k | σ 0 ) . 10 Pr o of. (i) Let ε = σ τ , where τ is an N × 1 standard normal uncertain vector, and σ = diag ( σ 0 , σ 0 , . . . , σ 0 ) is an N × N diagonal matrix. By Theorem 3 (Liu (2013)) on joint uncertain ty distributions, ε follo ws a multiv ariate normal distribution Φ ε ( x 1 , x 2 , · · · , x N ) =  N i =1 Φ ε i ( x i ; σ 0 ) , where Φ ε i ( x i ; σ 0 ) =  1 + exp  − π x i √ 3 σ 0  − 1 . Th us, ε ∼ N (0 , σ 0 I N ) . (ii) Denote e = X β = ( e 1 , e 2 , . . . , e N ) T . It is obvious that Z = X β + ε = e + ε . By Theorem 6 (Liu (2013)) on linear transformations of normal uncertain vectors, Z is a normal uncertain vector. F rom Theorem 3 (Liu (2013)) on join t uncertain ty distributions, Z follo ws a multiv ariate normal distributio n Φ Z ( x 1 , x 2 , · · · , x N ) =  N i =1 Φ z i ( x i ; e i , σ 0 ) , where Φ z i ( x i ; e i , σ 0 ) =  1 + exp  π ( e i − x i ) √ 3 σ 0  − 1 . Hence, Z ∼ N ( X β , σ 0 I N ) . (iii) F rom Theorem 5, we hav e ˆ β = ( X T X ) + X T Z − ( X T X ) + C T ˆ λ = QZ + P , where Q = ( X T X ) + X T = ( q ij ) p × N and P = − ( X T X ) + C T ˆ λ = ( P 1 , P 2 , . . . , P p ) T . F rom (ii), Z ∼ N ( X β , σ 0 I N ) , that is, Z = X β + σ 0 τ . Substituting yields that ˆ β = Q ( X β + σ 0 τ ) + P = ( QX β + P ) + σ 0 Q τ . By the constructiv e prop erty of Theorem 5, when Z = X β (i.e., τ = 0 ), w e hav e ˆ β = β . Hence, QX β + P = β , and ˆ β = β + σ 0 Q τ . By Theorem 6 (Liu (2013)) on linear transformations of normal uncertain vectors, ˆ β is still a normal uncertain vector. F or each comp onen t of ˆ β , ˆ β i = β i + σ 0  N j =1 q ij τ j . By the op erational law, ˆ β i ∼ N ( β i , N  j =1 | q ij | σ 0 ) . (16) Since ˆ β = ( ˆ β 1 , . . . , ˆ β p ) T = ( ˆ µ, ˆ a 1 , . . . , ˆ a r , ˆ b 1 , . . . , ˆ b s ) T , the distributions for ˆ µ , ˆ a i and ˆ b j follo w immediately . Based on Theorem 6, the α condence interv als are obtained as follows C I ( µ ) = ˆ µ ± σ µ √ 3 π ln  1 + α 1 − α  , (17) C I ( a i ) = ˆ a i ± σ a i √ 3 π ln  1 + α 1 − α  , C I ( b j ) = ˆ b j ± σ b j √ 3 π ln  1 + α 1 − α  , (18) where σ µ =  N j =1 | q 1 j | σ 0 , σ a i =  N j =1 | q i +1 ,j | σ 0 and σ b j =  N k =1 | q r + j +1 ,k | σ 0 . 11 Remark 4. In the two-factor exp eriment, we employ a single σ 0 for p ar ameter estimation, as in the single-factor c ase. Stil l, in the testing phase, we use individual σ i · or σ · j for e ach factor’s level, wher e σ i · 0 and σ · j 0 denote the standar d deviations c ompute d fr om the c ombine d samples of A i and B j , r esp e ctively. Next, w e ev aluate the equalit y of the eects under dieren t levels. F or the mo del (8), the h yp otheses are expressed as: H A 0 : a 1 = a 2 = · · · = a r , H A 1 : a 1 , a 2 , . . . , a r are not all equal , H B 0 : b 1 = b 2 = · · · = b s , H B 1 : b 1 , b 2 , . . . , b s are not all equal . Similar to the single-factor case, the homogeneit y test of a i (resp. b j ) is sucien t to determine whether all eects are zero. Before testing the homogeneit y of main eects, observ ations from the same factor level need to b e com bined. Sp ecically , to isolate the eect of factor A and p erform the homogeneit y test, observ ations at the same lev el of factor A across all levels of factor B are merged to form a complete sample set for testing, and similarly for factor B . F or instance, for the level i of factor A : z ik = z ij ( k − ∑ j − 1 l =1 m il ) , j − 1  l =1 m il < k ≤ j  l =1 m il , j = 1 , · · · , s, (19) and similarly for factor B as z j k . By collapsing the data by factor level, w e reduce the homogeneit y test for eac h main eect in a t wo-factor design to that of a single-factor eect. As describ ed in Section 2, we replace ˆ µ with a xed constant µ 0 , yielding ˜ z ik = z ik − µ 0 and ˇ z j k = z j k − µ 0 . The subsequent testing procedures are consistent with the single-factor case and will not b e rep eated here. 3.2 T w o-factor design with in teraction In practical exp erimen ts, the inuence of one factor may dep end on the level of another factor, leading to an interaction eect. In this subsection, we will establish an uncertain t wo-factor mo del with interaction to capture this non-additiv e relationship betw een factors. W e extend the mo del (8) by introducing the in teraction term AB . The weigh ts w ij = m ij / N represen t the prop ortion of total observ ations for the combination A i B j . F or an y balanced design, w ij = 1 /r s . The complete UFE mo del with interaction is dened:            z ij l = µ + a i + b j + ( ab ) ij + ε ij l , i = 1 , 2 , ..., r , j = 1 , 2 , ..., s, l = 1 , 2 , ..., m ij , r  i =1 w i · a i = 0 , s  j =1 w · j b j = 0 , r  i =1 w ij ( ab ) ij = 0 , s  j =1 w ij ( ab ) ij = 0 , ε ij l ∼ N (0 , σ 0 ) are m utually indep enden t . (20) Theorem 7. In the mo del (20) with a b alanc e d design, i.e., m ij = m ( i = 1 , . . . , r, j = 1 , . . . , s ) , let ˆ µ , ˆ a i , ˆ b j and ˆ ( ab ) ij b e the le ast-squar es estimators of µ , a i , b j and ( ab ) ij . Then, 12 the estimators ˆ µ , ˆ a i , ˆ b j ar e the same as those of (9)-(10) and ˆ ( ab ) ij = r s − r − s + 1 r sm m  l =1 z ij l + 1 − s r sm  k  = i m  l =1 z kj l + 1 − r r sm  n  = j m  l =1 z inl + 1 r sm  k  = i  n  = j m  l =1 z knl . Pr o of. Denote ˆ a = ( ˆ a 1 , ˆ a 2 , ..., ˆ a r ) , ˆ b = ( ˆ b 1 , ˆ b 2 , . . . , ˆ b s ) , ˆ ab = ( ˆ ( ab ) ij | i = 1 , 2 , . . . , r, j = 1 , 2 , . . . , s ) . By employing the least-squares metho d, ˆ µ , ˆ a i , ˆ b j and ˆ ( ab ) ij ( i = 1 , . . . , r , j = 1 , . . . , s ) are obtained b y solving ( ˆ µ, ˆ a , ˆ b , ˆ ab ) = argmin µ, a , b , ab r  i =1 s  j =1 m  l =1 ( z ij l − µ − a i − b j − ( ab ) ij ) 2 W e obtain the estimators ˆ µ , ˆ a i , ˆ b j and ˆ ( ab ) ij = 1 m m  l =1 z ij l − 1 sm s  j =1 m  l =1 z ij l − 1 r m r  i =1 m  l =1 z ij l + 1 r sm r  i =1 s  j =1 m  l =1 z ij l . Eac h estimator is written as a sum of indep endent normal uncertain v ariables, so that the op erational la w can b e applied to derive their uncertaint y distributions. Based on Theorem 7, it follo ws from the op erational law that ˆ ( ab ) ij ∼ N (( ab ) ij , 4(1 − 1 /r − 1 /s + 1 / ( r s )) σ 0 . F urther, the α condence interv al are (12), (13) and C I (( ab ) ij ) = ˆ ( ab ) ij ± 4  1 − 1 r − 1 s + 1 r s  √ 3 π σ 0 ln  1 + α 1 − α  . (21) The un balanced tw o-factor mo del with interaction is handled similarly to the unbal- anced case without in teraction using the Lagrangian m ultiplier metho d. The resp onse v ector Z and a ( N × p ) matrix X for p = 1 + r + s + r s are dened analogously , but β and x ij l include the in teraction terms: β =( µ, a 1 , . . . , a r , b 1 , . . . , b s , ( ab ) 11 , . . . , ( ab ) rs ) T , x ij l =(1 , 0 , . . . , 1 , . . . , 0 , 0 , . . . , 1 , . . . , 0 , 0 , . . . , 1 , . . . , 0) 1 × p suc h that x ij l β = µ + a i + b j + ( ab ) ij . The constrain t ((2 + r + s ) × p ) matrix C and d = 0 (2+ r + s ) × 1 are extended accordingly to enforce the tw o main-eect constraints and the additional r + s constrain ts on the in teraction terms, where C =   0 2 × 1 C a C b 0 2 × rs 0 r × 1 0 r × r 0 r × s C r ab 0 s × 1 0 s × r 0 s × s C s ab   , C a =  w 1 · · · · w r · 0 · · · 0  , C b =  0 · · · 0 w · 1 · · · w · s  , 13 and C r ab is an r × r s matrix with the ( i, ( i − 1) s + j ) th entry equal to w ij and zeros elsewhere, and C s ab is an s × r s matrix with the ( j, ( i − 1) s + j ) th entry equal to w ij and zeros elsewhere. The least-squares estimator ˆ β is obtained in exactly the same form as in the no- in teraction case (15). By the same argument as in Theorem 6, each comp onent satises (16). In particular, ˆ ( ab ) ij = ˆ β r + s +( i − 1) s + j +1 ∼ N  ( ab ) ij , N  k =1 | q r + s +( i − 1) s + j +1 ,k | σ 0  . The α condence interv als for µ , a i and b j retain the forms (17) and (18), while for the in teraction eect we hav e C I (( ab ) ij ) = ˆ ( ab ) ij ± σ ( ab ) ij √ 3 π ln  1 + α 1 − α  , (22) where σ ( ab ) ij =  N k =1 | q r + s +( i − 1) s + j +1 ,k | σ 0 . Note that q ij in the standard deviations of ˆ µ , ˆ a i , and ˆ b j dier b et w een the no-interaction and interaction mo dels b ecause of the expanded structure of X in Q = ( X T X ) + X T . The homogeneity tests for the main eects a i and b j pro ceed exactly as in the mo del without in teraction. F or testing the signicance of the in teraction, H AB 0 : ( ab ) ij = 0 ∀ i, j vs. H AB 1 : ∃ ( i, j ) s.t. ( ab ) ij 6 = 0 . (23) Similarly , the adjusted observ ations ˘ z ij l = z ij l − µ 0 − a i 0 − b j 0 preserv e indep endence across all com binations, where µ 0 , a i 0 , b j 0 are treated as kno wn xed constan ts. Theorem 8. F or hyp othesis pr oblem (23), the r eje ction r e gion at signic anc e level α is W =  i =1 ,...,r j =1 ,...,s W ij =  i =1 ,...,r j =1 ,...,s  ( ˘ z ij 1 , ˘ z ij 2 , ..., ˘ z ij m ij ) : ther e ar e at le ast α of indexes p ’s with 1 ≤ p ≤ m ij such that ˘ z ij p < Φ − 1 ( α/ 2) or ˘ z ij p > Φ − 1 (1 − α/ 2)  , wher e Φ − 1 ( α ) = σ ij 0 √ 3 π ln ( α 1 − α ) , and σ ij 0 ar e estimators of σ ij . Pr o of. Since µ 0 , a i 0 , and b j 0 are xed constants in the testing pro cedure, the adjustment ˘ z ij l preserv es mutual indep endence across dierent combinations ( i, j ) . The comp osite n ull hypothesis H AB 0 : ( ab ) ij = 0 , ∀ i, j can b e expressed as the intersection of individual h yp otheses H ij 0 : ( ab ) ij = 0 , i.e., H AB 0 =  i,j H ij 0 . Let W ij denote the rejection region of H ij 0 and W c ij b e the corresponding acceptance region. Then, the acceptance region for H AB 0 is  i,j W c ij . By De Morgan’s law, the rejection region for H AB 0 is W = ( ∩ i,j W c ij ) c = ∪ i,j W ij . This implies that H AB 0 is rejected if and only if ( ˘ z ij 1 , ˘ z ij 2 , . . . , ˘ z ij m ij ) fall in W ij for at least one com bination ( i, j ) . Since eac h W ij is an indep endent tw o-sided hypothesis test follo wing Y e and Liu (2022), the result follo ws. If H AB 0 is rejected, the in teraction eect of factors A and B is signicant at level α . 14 4 Real examples This section provides several examples to illustrate the detailed pro cesses of parameter esti- mation and hypothesis testing for uncertain single-factor and tw o-factor mo dels, including b oth balanced and unbalanced designs. Example 1. (A single-factor design) This study inv estigates the eect of caeine concen- tration on the transp ort of lab eled adenine across the blo o d-brain barrier from McCall et al. (1982) (T able 2). Let A b e a caeine concentration factor with three levels: A 1 = 0 . 1 mM, A 2 = 0 . 5 mM, and A 3 = 10 mM. T able 2: Data on the concen tration of lab eled adenine in the rat brains. i j z ij 1 1-5 2.91 4.14 6.29 4.40 3.77 2 1-4 5.80 5.84 3.18 3.18 3 1-6 3.05 1.94 1.23 3.45 1.61 4.32 In T able 2, z ij are the concen tration of lab eled adenine in the brain for the j th obser- v ation under level A i , µ i is the mean of the resp onse for lev el A i . T o analyze the eects of caeine concen tration, an uncertain m o del is constructed as follows: z ij = µ i + ε ij , ε ij ∼ N (0 , σ ) , or z ij = µ + a i + ε ij ,  3 i =1 a i = 0 with the ov erall mean µ and the xed eect a i of lev el i . Based on Theorem 1, w e obtain the estimators: ˆ µ 1 = 4 . 302 , ˆ µ 2 = 4 . 5 , and ˆ µ 3 = 2 . 6 . T able 3: The h yp othesis test of residuals. ε i · e i 0 σ i 0 AI ( ε i · ; e i 0 , σ i 0 ) Singular P oint ε 1 · 0 1.114 [-2.251, 2.251] 0 ε 2 · 0 1.320 [-2.666, 2.666] 0 ε 3 · 0 1.094 [-2.209, 2.209] 0 W e test whether the residuals ˆ ε ij = z ij − ˆ µ i follo w N (0 , σ i ) . T able 3 shows that each residual ε i ∼ N (0 , σ i 0 ) . Then, consider the follo wing h yp otheses: H 0 : σ 1 = σ 2 = σ 3 v ersus σ 1 , σ 2 , σ 3 are not all equal . The rejection region of H 0 at the signicance lev el α = 0 . 05 is W =  ( ε i 1 , ε i 2 , . . . , ε im i ) : ∃ i 6 = j ( i, j ∈ { 1 , 2 , 3 } ) such that at least α of indexes p ’s with 1 ≤ p ≤ m i satisfy ε ip < Φ − 1 ( α/ 2; σ j 0 ) or ε ip > Φ − 1 (1 − α/ 2; σ j 0 )  , 15 where Φ − 1 ( α ; σ j 0 ) = σ j 0 √ 3 π ln ( α 1 − α ) . T able 3 reveals that all ε i · fall in AI ( ε i · ; σ j 0 )( i 6 = j, i, j = 1 , 2 , 3) , H 0 can not b e rejected. That is to say , there is no signican t dierence in the stan- dard deviations of ε i ( i = 1 , 2 , 3) at the signicance level 0.05. Thus, the residuals satisfy the assumption of homogeneit y of standard deviations. F urther, an uncertain common test can b e p erformed to determine whether the standard deviations σ 1 = σ 2 = σ 3 = σ are equal to a xed constan t σ 0 . F or this, w e consider the hypotheses H ∗ 0 : σ = σ 0 v ersus H ∗ 1 : σ 6 = σ 0 . The 3 sets of residuals are then combined in to one set, i.e., { ε 1 , ε 2 , ..., ε 15 } = { ε 11 , ..., ε 15 , ε 21 , ..., ε 24 , ε 31 , ..., ε 36 } , and σ 2 0 = 1 15  15 i =1 ( ε i ) 2 = 1 . 165 2 . Giv en a signicance lev el α = 0 . 05 , Φ − 1 ( α/ 2; σ 0 ) = − 2 . 353 , Φ − 1 (1 − α/ 2; σ 0 ) = 2 . 353 . Since α × 15 = 0 . 75 , w e hav e W =  ( ε 1 , ε 2 , . . . , ε 15 ) : there are at least 1 of indexes p ’s with 1 ≤ p ≤ 15 such that ε p < − 2 . 353 or ε p > 2 . 353  . Note that all the ε j fall in [ − 2 . 353 , 2 . 353] . Th us, ( ε 1 , ε 2 , . . . , ε 15 ) / ∈ W , and H ∗ 0 cannot b e rejected. Then, the standard deviation σ is equal to a xed constan t σ 0 = 1 . 165 . Under the indep endence assumption, we can obtain the 95% uncertain condence in- terv al of µ i ( i = 1 , 2 , 3) : 4 . 302 ± 2 . 353 , 4 . 5 ± 2 . 353 , and 2 . 6 ± 2 . 353 from (2). Next, we test the homogeneit y of means: H a : µ 1 = µ 2 = µ 3 v ersus H b : µ 1 , µ 2 , µ 3 are not all equal . The rejection region of H a at the signicance lev el α = 0 . 05 is W =  ( z i 1 , z i 2 , . . . , z im i ) : ∃ i 6 = j ( i, j ∈ { 1 , 2 , 3 } ) such that at least α of indexes p ’s with 1 ≤ p ≤ m i satisfy z ip < Φ − 1 ( α/ 2; µ j 0 ) or z ip > Φ − 1 (1 − α/ 2; µ j 0 )  , where Φ − 1 ( α ; µ j 0 ) = µ j 0 + σ i 0 √ 3 π ln ( α 1 − α ) . T able 4 shows that z 3 j ( j = 2 , 3 , 5) are not in AI ( z 3 · ; µ i 0 )( i = 1 , 2) , z 1 j ( j = 3) not in AI ( z 1 · ; µ 30 ) , and z 2 j ( j = 1 , 2) not in AI ( z 2 · ; µ 30 ) . It means that there is a signicant dierence in the means of A 3 and A i ( i = 1 , 2) at the signicance lev el 0.05. Thus, H a should b e rejected. F rom (4)–(6), the 95% uncertain condence in terv al of µ and a i ( i = 1 , 2 , 3) satisfy: 3 . 674 ± 2 . 353 , 0 . 628 ± 3 . 137 , 0 . 826 ± 3 . 451 , and − 1 . 074 ± 2 . 824 . Based on the estimators, the uncertain homogeneit y test of eects can b e conducted H 0 : a 1 = a 2 = a 3 v ersus H 1 : a 1 , a 2 , a 3 are not all equal . Using Theorem 3, we obtain the results sho wn in T able 4, where ˜ z 3 j ( j = 2 , 3 , 5) are not in AI ( ˜ z 3 · ; a i 0 ) ( i = 1 , 2) , ˜ z 1 j ( j = 3) is not in AI ( ˜ z 1 · ; a 30 ) , and ˜ z 2 j ( j = 1 , 2) are not in AI ( ˜ z 2 · ; a 30 ) . Th us, there is a signicant dierence in the eects of A 3 and A i ( i = 1 , 2) at the signicance lev el 0.05. It shows that H 0 should b e rejected. 16 T able 4: The uncertain homogeneit y test. z i · or ˜ z i · i = 1 i = 2 i = 3 AI ( z i · ; µ 10 ) [2.051, 6.553] [1.636, 6.968] [2.093, 6.511] AI ( z i · ; µ 20 ) [2.249, 6.751] [1.834, 7.166] [2.291, 6.709] AI ( z i · ; µ 30 ) [0.349, 4.851] [-0.066, 5.266] [0.391, 4.809] AI ( ˜ z i · ; a 10 ) [-1.622, 2.878] [-2.032, 3.288] [-1.582, 2.838] AI ( ˜ z i · ; a 20 ) [-1.424, 3.076] [-1.834, 3.486] [-1.384, 3.036] AI ( ˜ z i · ; a 30 ) [-3.324, 1.176] [-3.734, 1.586] [-3.284, 1.136] In conclusion, caeine concentration signicantly aects the transp ort of lab eled ade- nine across the blo o d-brain barrier. Statistical analysis also rev eals a signicant dierence b et w een the third concentration lev el ( A 3 ) and the other t w o ( A 1 and A 2 ). The conclusion dra wn from Example 1 is consistent with that obtained under the probability framework (McCall et al. (1982)). Example 2. (A two-factor b alanc e d design) This study in v estigates the eects of target densit y and the fraction of slash chips on the actual density of particleb oards, as rep orted b y Bo ehner (1975) (T able 5). The t wo factors are target densit y A and the fraction of slash c hips B . T arget densit y has t w o levels: A 1 = 42 lb/ft 3 and A 2 = 48 lb/ft 3 . The fraction of slash c hips also has tw o levels: B 1 = 0% and B 2 = 25% . T able 5: Data on the densit y of particleb oards. i j l z ij l 1 1 1-3 40.9 42.8 45.4 2 1-3 41.9 43.9 46.0 2 1 1-3 44.4 48.2 49.9 2 1-3 46.2 48.6 50.8 In T able 5, z ij l represen ts the actual density of the l th b oard under A i B j . T o analyze these factors, a balanced t w o-factor mo del with in teraction is constructed as follo ws z ij l = µ + a i + b j + ( ab ) ij + ε ij l , ε ij l ∼ N (0 , σ ) , and  2 i =1 w i · a i = 0 ,  2 j =1 w · j b j = 0 ,  2 i =1 w ij ( ab ) ij = 0 ,  2 j =1 w ij ( ab ) ij = 0 , where µ is the ov erall mean, a i is the main eect of A i , b j is the main eect of B j , and ( ab ) ij is the in teraction eect b et ween these tw o factor levels. Using Theorem 7, we obtain the estimators: ˆ µ = 45 . 75 , ˆ a 1 = − 2 . 267 , ˆ a 2 = 2 . 267 , ˆ b 1 = − 0 . 483 , ˆ b 2 = 0 . 483 , ˆ ( ab ) 11 = 0 . 033 , ˆ ( ab ) 12 = − 0 . 033 , ˆ ( ab ) 21 = − 0 . 033 , ˆ ( ab ) 22 = 0 . 033 . W e then test whether the residuals ˆ ε ij l = z ij l − ˆ µ − ˆ a i − ˆ b j − ˆ ( ab ) ij follo w N (0 , σ ij ) . T able 6 sho ws that each residual ε ij ∼ N (0 , σ ij 0 ) . Next, w e consider the follo wing hypotheses: H 0 : σ 11 = σ 12 = σ 21 = σ 22 v ersus H 1 : σ 11 , σ 12 , σ 21 , σ 22 are not all equal . 17 The rejection region of H 0 at the signicance level α = 0 . 05 is similar to Example 1. T able 6 rev eals that all ε ij · fall in AI ( ε ij · ; σ uv 0 )( ij 6 = uv , ij, uv ∈ { 11 , 12 , 21 , 22 } ) , and H 0 can not b e rejected. The residuals satisfy the assumption of homogeneity of standard deviations. T able 6: Results of the h yp othesis test for residuals. ε ij · e ij 0 σ ij 0 AI ( ε ij · ; e ij 0 , σ ij 0 ) Singular P oint ε 11 · 0 1.845 [-3.725, 3.725] 0 ε 12 · 0 1.674 [-3.381, 3.381] 0 ε 21 · 0 2.299 [-4.644, 4.644] 0 ε 22 · 0 1.879 [-3.794, 3.794] 0 Moreo ver, the uncertain common test is conducted to determine whether the standard deviations σ 11 = σ 12 = σ 21 = σ 22 = σ are equal to a xed constant σ 0 . Based on this, w e consider the follo wing hypotheses: H ∗ 0 : σ = σ 0 v ersus H ∗ 1 : σ 6 = σ 0 . It is conducted in the same manner as in Example 1. All the ε j fall in [ − 3 . 914 , 3 . 914] . Th us, ( ε 1 , ε 2 , . . . , ε 12 ) / ∈ W , and H ∗ 0 cannot b e rejected. It means that the standard deviation σ 0 = 1 . 938 . Through (12)-(13) and (21), the 95% uncertain condence in terv al of µ , a i , b j , and ( ab ) ij are: µ = 45 . 75 ± 3 . 915 , a 1 = − 2 . 267 ± 3 . 915 , a 2 = 2 . 267 ± 3 . 915 , b 1 = − 0 . 483 ± 3 . 915 , b 2 = 0 . 483 ± 3 . 915 , ( ab ) 11 = 0 . 033 ± 3 . 915 , ( ab ) 12 = − 0 . 033 ± 3 . 915 , ( ab ) 21 = − 0 . 033 ± 3 . 915 , ( ab ) 22 = 0 . 033 ± 3 . 915 . After that, w e pro ceed with the homogeneity test of the main eects and the signicance test of the in teraction eect. The data corresponding to factor A are com bined as (19), and similarly for factor B . z ik and z j k are com bined data, where i = 1 , 2 represen ts the levels of factor A , j = 1 , 2 represents the lev els of factor B , and k denotes the index ranging from 1 to m i for i and from 1 to m j for j . This combination ensures that all relev an t data is organized for subsequen t tests. W e then consider the h yp otheses (23) and H A 0 : a 1 = a 2 v ersus H A 1 : a 1 6 = a 2 , H B 0 : b 1 = b 2 v ersus H B 1 : b 1 6 = b 2 . After collapsing the observ ations by factor level, the rejection regions for testing H A 0 and H B 0 are constructed according to Theorem 3. In T able 7, ˜ z 1 k ( k = 1 , 2 , 4 , 5) not in AI ( ˜ z 1 · ; a 20 ) and ˜ z 2 k ( k = 2 , 3 , 5 , 6) not in AI ( ˜ z 2 · ; a 10 ) . Thus, H A 0 is rejected, indicating that factor A is signicant at the 0.05 lev el. In contrast, all ˇ z j · fall in AI ( ˇ z j · ; b i 0 ) , so H B 0 cannot b e rejected, indicating that factor B is not signicant at the 0.05 level. Using Theorem 8, we obtain that ˘ z 11 l fall in [-3.727, 3.727], ˘ z 12 l fall in [-3.381, 3.381], ˘ z 21 l fall in [-4.644, 4.644], and ˘ z 22 l fall in [-3.795, 3.795]. It means that H AB 0 can not b e rejected, that is, the in teraction eect b etw een the factors A and B are not signicant. In conclusion, target density has a signicant impact on particleb oard actual densit y , whereas the prop ortion of slash c hip (0% and 25%) has no signican t eect. No in teraction w as found b etw een these factors. The results of Example 2 are consistent with those obtained under the probabilit y framework (Bo ehner (1975)). 18 T able 7: Results of the homogeneit y tests for main eects. ˜ z i · or ˇ z j · i or j = 1 i or j = 2 AI ( ˜ z i · ; a 10 ) [-5.939, 1.405] [-6.634, 2.100] AI ( ˜ z i · ; a 20 ) [-1.405, 5.939] [-2.100, 6.634] AI ( ˇ z j · ; b 10 ) [-6.976, 6.010] [-6.724, 5.757] AI ( ˇ z j · ; b 20 ) [-6.010, 6.976] [-5.757, 6.724] Example 3. (A two-factor unb alanc e d design) Consider the same tw o-factor interaction mo del as Example 2, but with an un balanced design. The data is from Timm (1998) (T able 8). Eac h factor ( A and B ) has t wo lev els. Since the analysis follo ws the same procedures as in Example 2, the detailed steps are omitted here. The results are presen ted in T ables 9–10. A t the signicance level 0.05, the normality and homogeneity of the standard deviations assumptions are satised for residuals. Based on the uncertain common test, we hav e σ 0 = 7 . 429 . F rom (17)–(18) and (22), the 95% uncertain condence interv al of µ , a i , b j , and ( ab ) ij are: µ = 56 . 818 ± 6 . 669 , a 1 = 11 . 852 ± 10 . 003 , a 2 = − 14 . 222 ± 10 . 003 , b 1 = − 2 . 040 ± 10 . 003 , b 2 = 1 . 700 ± 10 . 003 , ( ab ) 11 = − 4 . 630 ± 15 . 005 , ( ab ) 12 = 4 . 630 ± 15 . 005 , ( ab ) 21 = 6 . 944 ± 15 . 005 , ( ab ) 22 = − 4 . 630 ± 15 . 005 . The homogeneity tests indicate that the main eects of factor A are signicant, whereas those of factor B are not. Moreov er, the in teraction eect is signicant, as ˘ z 212 lies outside the in terv al [ − 11 . 109 , 11 . 109] . T able 8: Un balanced data. i j l z ij l 1 1 1-3 61 73 52 2 1-3 79 65 81 2 1 1-2 42 53 2 1-3 37 32 50 T able 9: Results of the h yp othesis test for residuals. ε ij · e ij 0 σ ij 0 AI ( ε ij · ; e ij 0 , σ ij 0 ) Singular P oint ε 11 · 0 8.602 [-17.375, 17.375] 0 ε 12 · 0 7.118 [-14.377, 14.377] 0 ε 21 · 0 5.500 [-11.109, 11.109] 0 ε 22 · 0 7.587 [-15.323, 15.323] 0 Through calculating the exp ected resp onse v alues for each combination, we determine that, if the exp erimental objective is “larger-the-better”, the optimal combination is A 1 B 2 , with an exp ected resp onse v alue of 75.000; if the exp erimental ob jectiv e is “smaller-the- b etter”, the optimal combination is A 2 B 2 , with an exp ected resp onse v alue of 39.666. 19 T able 10: Results of the homogeneit y tests for main eects. ˜ z i · or ˇ z j · i or j = 1 i or j = 2 AI ( ˜ z i · ; a 10 ) [-8.804, 32.508] [-3.969, 27.673] AI ( ˜ z i · ; a 20 ) [-34.878, 6.434] [-30.043, 1.599] AI ( ˇ z j · ; b 10 ) [-22.929, 18.849] [-40.693, 36.613] AI ( ˇ z j · ; b 20 ) [-19.189, 22.589] [-36.953, 40.353] Although the main eect of factor B is not signicant, its interaction with factor A is sig- nican t, so the choice of factor B ’s level still has an important impact on the results across dieren t levels of factor A . Unlik e the analysis under the probability framework (Timm (1998)), which failed to detect a signicant in teraction, the uncertain mo del iden tied a signican t in teraction eect in this unbalanced design. This demonstrates the p otential of the uncertain approach to reveal underlying eects that migh t b e obscured in traditional probabilistic analysis, esp ecially with small samples. 5 Conclusion Within the framework of uncertaint y theory , this pap er conducted an in-depth analysis of uncertain xed-eects mo dels using single- and tw o-factor designs. Under UFE mo dels, the study pro vides a comprehensive solution for handling uncertain exp erimental data. These metho dologies not only eectively address the imbalance in single-factor exp eri- men tal designs but also address interaction eect and unequal replication in t w o-factor designs. Regarding the parameter estimation and testing problem, several critical tech- nical c hallenges ha v e b een ov ercome. F or t w o-factor unbalanced designs, we ingeniously transformed the model in to a multiv ariate linear regression mo del com bined with uncertain v ectors to obtain the parameter distributions, successfully resolving the limitations of tra- ditional metho ds that could not deriv e explicit solutions. In the testing phase, w e designed a nov el data adjustmen t technique. By utilizing xed constants, which can b e estimated from data but are treated as kno wn, to mo dify the data, we eectively maintained inter- group indep endence. More imp ortan tly , this study addresses a fundamental challenge in exp erimen tal design: the conserv atism of probabilistic metho ds in small sample settings. The prop osed UFE mo dels, ro oted in uncertaint y theory , ov ercome limitations such as lo w p o w er b y not relying on large-sample asymptotics. Their thic k-tailed distributions and test logic based on all data p oints yield a stricter, more robust inference that reliably captures true dierences often obscured in traditional analyses. By analyzing three designs, w e v ali- dated the eectiv eness and practicality of the proposed methods. These cases encompassed single-factor unbalanced designs, tw o-factor balanced designs, and t w o-factor unbalanced designs, comprehensively demonstrating the adaptability and exibility of the prop osed metho dologies across v arious scenarios. Our prop osed mo del and metho ds can b e extended to other t yp es of designs, such as randomized blo c k, split-plot, and orthogonal designs. F uture inv estigations could fo cus on 20 applying these metho dologies. Conict of in terest The authors declare that they ha v e no conict of in terest. 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