1 Introduction

In complex real-world studies, uncertain data can arise due to noise or incomplete information. Consider, for example, a 3 × 3 balanced lattice design experiment where nine different diets were compared with four replications to determine their impact on non-protein nitrogen in rat colostrum. During the experiment, it is likely that we will encounter uncertain data. However, handling this data properly is crucial for accurate statistical analysis. If not addressed, it could introduce bias in treatment effect estimation, inflate the sum of squares, or increase estimate variability. Therefore, this problem should be carefully treated.

From the first beginning, the lattice design was introduced by [1] as a quasi-factorial design, which is characterized by the fact that the number of treatments is a perfect square, and the size of the blocks is the square root of the number of treatments. It should be noted that incomplete blocks are combined into groups to form separate replications. The number of replications of the treatments is also flexible and is helpful in situations where many treatments are to be tested. In this context, an analysis of simple lattice designs with unequal replications was first presented by [2]. Brenna and Kramer [3] discussed the use of factorial treatment combinations in rectangular lattice designs. Generalized lattice square designs were provided by [4]. Several formulae are given by [5] for analyzing square and rectangular lattice designs. Williams, et al. [6] presented a general definition of lattice square designs and a unified method for analyzing restricted lattice square designs. A thesis was written by [7] on the analysis of square lattice designs using R and SAS. A PC-SAS program for analyzing data obtained from a lattice design was presented by [8]. The "agricola" Package was developed to analyze lattice design data [9]. You can find related articles and books about lattice design [10,11,12]. As mentioned above, balanced lattice designs under classical statistics have been extensively studied and utilized. However, there is a notable gap in their application within environments characterized by uncertainty.

In uncertain data frameworks and studies, Jan, et al. [13] proposed a new method for analyzing complex and unreliable data by combining bipolar complex fuzzy set and soft set theories into bipolar complex fuzzy soft relations which allow for a dual-direction membership. Also, Mikhaylov, et al. [14] examined the relationship between oil prices and the exchange rates of oil-exporting countries, employing a fuzzy decision-making model by considering the bipolar model that incorporates expert opinions and econometric analysis. On the another hand and under observation error, Stratton, et al. [15] introduced a Bayesian joint spatial modeling method to assess the impact of white-nose syndrome in observing the bat populations in North America.

Since where are adopting the neutrosophic approach, we will shed some light on the brief history of this statistical branch. The concept of Neutrosophic Statistics (NS) has been introduced by [16] to deal with data whose neutrosophic numbers have some degree of indeterminacy. The differences between fuzzy statistics, NS, and classical statistics are explained well enough in [17]. Also, Aslam [18] has proposed and defend the neutrosophic ANOVA. A more recent article discussed the application of neutrosophic analysis of covariance to neutrosophic completely random designs, neutrosophic randomized complete block designs, and neutrosophic split-plot designs [19]. AlAita, et al. [20] introduced a generalized approach that utilizes neutrosophic statistics to enhance the analysis of split-plot and split-block designs within uncertain environments. AlAita and Talebi [21] presented a neutrosophic augmented randomized complete block design to address the challenges of missing control treatments and uncertainties in data within plant breeding programs. Besides, post hoc multiple comparison tests under NS have been proposed by [22]. Aslam and Al-Marshadi [23] introduced the Watson-Williams test under NS for the analysis of circular and angular data that is uncertain, imprecise, and indeterminate. In another paper, diagnosis test under NS have been discussed by [24]. Numerous statistical tests have been discussed under NS in [25,26,27,28,29,30,31,32].

In our exploration of balanced lattice designs, we raised some ambiguities in data, such as indeterminacy and the imprecision of the F-test. The apparent limitations of traditional statistical methods in managing these complexities have led us to embrace NS. This paper explores the resolution of indeterminacies by calculating ANOVA Table within the NS framework and deriving an F-test that provides more precise understanding of uncertainty. Notably, this is the first discussion of balanced lattice designs in a neutrosophic context, marking a significant advancement in the field.

The remainder of this paper is arranged as follows: We will discuss some concepts under neutrosophic statistics in the following Section. We will provide the statistical analysis of the neutrosophic balanced lattice design in Section 3 in Section 4, detailed numerical examples will be provided. Section 5 will be discussed the advantages of a neutrosophic balanced lattice design. Future research directions and recommendations are given in Section 6. Finally, the conclusion will be presented in Section 7.

2 Preliminaries

In recent years, many studies have discussed using neutrosophic data (where has some degree of indeterminacy) in many real-world applications. A significant problem may arise in the statistical analysis of indeterminate data that is not appropriately handled. To solve this problem, neutrosophic statistics were defined and used as a generalization of classical statistics. The following is a brief on some basic concepts related to NS.

Suppose that a neutrosophic random variable (NRV) \({Y}_{N}\in \left[{Y}_{L}, {Y}_{U}\right]\) follows the neutrosophic normal distribution (NND) with a neutrosophic population mean \({\mu }_{N}\in \left[{\mu }_{L}, {\mu }_{U}\right]\) and a neutrosophic population variance \({\sigma }_{N}^{2}\in [{\sigma }_{L}^{2},{\sigma }_{U}^{2}]\), where \({Y}_{L}\) and \({Y}_{U}\) are smaller and larger values of indeterminacy interval. Let \({Y}_{N}= {Y}_{L}+ {Y}_{U}{I}_{N}\) be the neutrosophic form of NRV having determinate part \({Y}_{L}\) and indeterminate part \({Y}_{U}{I}_{N}\); \({I}_{N}\in [{I}_{L},{I}_{U}]\), where \({I}_{N}\in [{I}_{L},{I}_{U}]\) is indeterminacy interval.

Suppose \({n}_{N}\in \left[{n}_{L}, {n}_{U}\right]\) be a neutrosophic random sample selected from a population of size \({N}_{N}\) having indeterminate observations. The neutrosophic population means \({\mu }_{N}\) and the variance \({\sigma }_{N}^{2}\), are expressed as follows;

$${\mu }_{N}\epsilon \left[\frac{{\sum }_{i=1}^{{N}_{L}}{Y}_{Li}}{{N}_{L}},\frac{{\sum }_{i=1}^{{N}_{U}}{Y}_{Ui}}{{N}_{U}}\right];\;{\mu }_{N}\epsilon \left[{\mu }_{L},{\mu }_{U}\right]$$

and

$${\sigma }_{N}^{2}\in \left[\frac{\sum_{i=1}^{{N}_{L}}{\left({Y}_{Li}-{\mu }_{L}\right)}^{2}}{{N}_{L}},\frac{\sum_{i=1}^{{N}_{U}}{\left({Y}_{Ui}-{\mu }_{U}\right)}^{2}}{{N}_{U}}\right];\;{\sigma }_{N}^{2}\in [{\sigma }_{L}^{2},{\sigma }_{U}^{2}]$$

But, in the numerical examples, \({\mu }_{N}\) and \({\sigma }_{N}^{2}\) are unknown and can be estimated using the sample observations. The neutrosophic sample mean \({\overline{Y} }_{N}\) and the variance \({s}_{N}^{2}\), are expressed by;

$${\overline{Y} }_{N}\in \left[\frac{\sum_{i=1}^{{n}_{L}}{Y}_{Li}}{{n}_{L}},\frac{\sum_{i=1}^{{n}_{U}}{Y}_{Ui}}{{n}_{U}}\right];\;{\overline{Y} }_{N}\in \left[{\overline{Y} }_{L}, {\overline{Y} }_{U}\right]$$

and

$${s}_{N}^{2}\in \left[\frac{\sum_{i=1}^{{n}_{L}}{\left({Y}_{Li}-{\overline{Y} }_{L}\right)}^{2}}{{n}_{L}-1},\frac{\sum_{i=1}^{{n}_{U}}{\left({Y}_{Ui}-{\overline{Y} }_{U}\right)}^{2}}{{n}_{U}-1}\right];\;{s}_{N}^{2}\in \left[{s}_{L}^{2},{s}_{U}^{2}\right],$$

where \({\overline{Y} }_{L}=\frac{\sum_{i=1}^{{n}_{L}}{Y}_{Li}}{{n}_{L}}\) and \({\overline{Y} }_{U}=\frac{\sum_{i=1}^{{n}_{U}}{Y}_{Ui}}{{n}_{U}}\) are the smaller and larger values of the indeterminacy interval, respectively.

To provide a brief comparison of fuzzy statistics, NS, and classical statistics, we can list the following differences:

Classical statistics

Fuzzy statistics

Neutrosophic statistics

Classical statistics require all observations or parameters to be precisely determined before they can be analyzed

Fuzzy statistics consist of observations or parameters that are fuzzy, and that are not taking account of the measure of uncertainty

Neutrosophic statistics are regard as a measure of indeterminacy and are extension of fuzzy statistics. In an uncertain environment, NS reduces to classical statistics when all observations or parameters have been determined

3 Statistical Analysis of the Proposed Design

Under NS, let \({t}_{N}\) represents the total number of neutrosophic treatments, \({k}_{N}\) represents neutrosophic block size, \({s}_{N}\) represents the number of neutrosophic blocks per neutrosophic replication which is equal to \({k}_{N}\), \({r}_{N}\) represents the number of neutrosophic replications (for neutrosophic balanced designs, \({r}_{N}= {k}_{N}+1\)). Let \({y}_{Nij(l)}\) represents the response value of the jth neutrosophic treatment in the lth block within ith neutrosophic replication, \(i = \text{1,2},\dots ,{k}_{N}+ 1\), \(j = \text{1,2},\dots {k}_{N}^{2}\), \(l = \text{1,2},\dots ,{r}_{N}{k}_{N}\). The neutrosophic statistical model is.

$${y}_{Nij(l)}={\mu }_{N}+{\alpha }_{Ni}+{\beta }_{Ni(l)}+{\tau }_{Nj}+{\varepsilon }_{Nij(l)},$$

where \({\mu }_{N}\), \({\alpha }_{Ni}\), \({\beta }_{Ni(l)}\), and \({\tau }_{Nj}\) stands for the effect of the neutrosophic mean, the neutrosophic replicate, the neutrosophic incomplete block, and the neutrosophic treatment, respectively. \({\varepsilon }_{Nij(l)}\) stands for the intra-block residual, assumed to be normally and independently distributed with mean zero and variance \({\sigma }_{N}^{2}\in [{\sigma }_{L}^{2},{\sigma }_{U}^{2}]\).

Randomization

We demonstrate the randomization process of the proposed design using a field experiment involving sixteen neutrosophic treatments. Under NS, layout of the design involves the following steps:

  • Step 1: The number of replications required is determined by dividing the experimental area into \({r}_{N}=({k}_{N} + 1)\) such that each replication has \({t}_{N}= {k}_{N}^{2}\) experimental plots.

  • Step 2: Each replication should be divided into \({k}_{N}\) incomplete blocks. This example would involve four incomplete blocks, each containing four experimental plots, as illustrated in Fig. 1.

Fig. 1
figure 1

Division of the experimental area in a neutrosophic balanced design for sixteen treatments in blocks of four units

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  • Step 3: We choose a basic plan based on the number of neutrosophic treatments that will be tested. The neutrosophic balanced lattice design selected for the present study is shown in Table 1.

    Table 1 basic plan for a 4 × 4 neutrosophic balanced lattice design for sixteen treatments in blocks of four units
    Full size table

  • Step 4: Using the completely randomized design (CRD), we randomize the replication arrangements, the incomplete blocks within each replication, and the neutrosophic treatment arrangements within each block of the selected plan for field layout. As shown in Table 2, the neutrosophic replications, blocks, and treatments have been reorganized.

    Table 2 A selected basic plan with rearranged neutrosophic replications, blocks, and treatments in a neutrosophic balanced lattice design
    Full size table

  • Step 5: The rearranged plan is used to set up the experiment in the field. According to Fig. 2, every neutrosophic treatment appears in each of the four rows and every pair of neutrosophic treatments appears once in the same block. By using such a design, it is possible to compare each pair of neutrosophic treatments with equal precision.

    Fig. 2
    figure 2

    Allocation of sixteen neutrosophic treatments in a field layout with four neutrosophic blocks

    Full size image

The formulas for NANOVA sums of squares can be expressed as follows:

  1. 1.

    Neutrosophic total sum of squares is.

    \({SS}_{NT}=\sum_{i=1}^{{r}_{N}}\sum_{j=1}^{{k}_{N}^{2}}{y}_{Nij(l)}^{2}-{CF}_{N}\); \({SS}_{NT}\in \left[{SS}_{LT},{SS}_{UT}\right]\), where \({CF}_{N}=\frac{{y}_{N\dots }^{2}}{{r}_{N}{k}_{N}^{2}}\).

  2. 2.

    Unadjusted neutrosophic treatment sum of squares is

    $$\begin{array}{ccc}{SS}_{NTr(unadj)}=\frac{1}{{r}_{N}}\sum_{k=1}^{{k}_{N}^{2}}{T}_{Nj}^{2}-{CF}_{N}& ;& {SS}_{NTr(unadj)}\in \left[{SS}_{LTr(unadj)},{SS}_{UTr(unadj)}\right]\end{array}$$

    where \({T}_{Nj}\) is the sum of observations for neutrosophic treatment \(j\).

  3. 3.

    Neutrosophic replication sum of squares is

    $${SS}_{NR}=\frac{1}{{k}_{N}^{2}}\sum\nolimits_{i=1}^{{r}_{N}}{R}_{Ni}^{2}-{CF}_{N};\;{SS}_{NR}\in \left[{SS}_{LR},{SS}_{UR}\right]$$

    where \({R}_{Ni}\) is the sum of observations in neutrosophic replication \(i\).

  4. 4.

    To calculate the adjusted neutrosophic block sum of squares, \({SS}_{NB(adj)}\), a number of quantities must first be calculated. Let \({B}_{Nj}\) stands for the sum of block totals for the neutrosophic blocks with neutrosophic treatment \(j\), \(j = \text{1,2},\dots {k}_{N}^{2}\), \({W}_{Nj}\) stands for the weight for the \(j\) th neutrosophic treatment which is used for adjustment for neutrosophic block,

    $${W}_{Nj}={k}_{N}{T}_{Nj}-\left({k}_{N}+1\right){B}_{Nj}+{G}_{N},$$

    where \({G}_{N}=\sum_{i=1}^{{r}_{N}}\sum_{j=1}^{{k}_{N}^{2}}{y}_{Nij(l)}\), and \(\sum_{j=1}^{{k}_{N}^{2}}{W}_{Nj}=0\). Thus, adjusted neutrosophic block sum of squares is

    $${SS}_{NB(adj)}=\frac{\sum_{j=1}^{{k}_{N}^{2}}{W}_{Nj}^{2}}{{k}_{N}^{2}\left({k}_{N}+1\right)};\;{SS}_{NB(adj)}\in \left[{SS}_{LB(adj)},{SS}_{UB(adj)}\right]$$
  5. 5.

    Neutrosophic intra-block error sum of squares is

    $${SS}_{NE}={SS}_{NT}-{SS}_{NTr\left(unadj\right)}-{SS}_{NR}-{SS}_{NB\left(adj\right)};{SS}_{NE}\in \left[{SS}_{LE},{SS}_{UE}\right]$$

Table 3 presents the NANOVA for neutrosophic balanced lattice design.

Table 3 NANOVA Table for neutrosophic balanced lattice design
Full size table

The \({MS}_{NTr\left(unadj\right)}\) of the unadjusted treatment cannot be used to test against the \({MS}_{NE}\) of intra-block error since the \({MS}_{NTr\left(unadj\right)}\) of the unadjusted neutrosophic treatment still contains neutrosophic block effects. The adjusted neutrosophic treatment \({SS}_{NTr\left(adj\right)}\) is defined as

$${SS}_{NTr\left(adj\right)}=\frac{\sum_{j=1}^{{k}_{N}^{2}}{\left({T}_{Nj}+{\mu }_{N}{W}_{Nj}\right)}^{2}}{{r}_{N}}-{CF}_{N}=\frac{\sum_{j=1}^{{k}_{N}^{2}}{\left({T}_{Nj}{^\prime}\right)}^{2}}{{r}_{N}}-{CF}_{N};\;{SS}_{NTr\left(adj\right)}\in \left[{SS}_{LTr\left(adj\right)},{SS}_{UTr\left(adj\right)}\right]$$

where \({T}_{Nj}{^\prime}\) represents the adjusted neutrosophic treatment total, and \({\mu }_{N}\) represents the adjustment neutrosophic factor for error and neutrosophic treatment means defined as.

$${\mu }_{N}=\frac{{MS}_{NB\left(adj\right)}-{MS}_{NE}}{{k}_{N}^{2}{MS}_{NB\left(adj\right)}}$$

Neutrosophic mean squares NMS are defined as:

$${MS}_{NB(adj)}=\frac{{SS}_{NB(adj)}}{{k}_{N}^{2}-1};{MS}_{NB(adj)}\in \left[{MS}_{LB(adj)},{MS}_{UB(adj)}\right]$$
$${MS}_{NTr\left(unadj\right)}=\frac{{SS}_{NTr\left(unadj\right)}}{{k}_{N}^{2}-1}; {MS}_{NTr\left(unadj\right)}\in \left[{MS}_{LTr\left(unadj\right)},{MS}_{UTr\left(unadj\right)}\right]$$
$${MS}_{NTr\left(adj\right)}=\frac{{SS}_{NTr\left(adj\right)}}{{k}_{N}^{2}-1}; {MS}_{NTr\left(adj\right)}\in \left[{MS}_{LTr\left(adj\right)},{MS}_{UTr\left(adj\right)}\right]$$
$${MS}_{NE}=\frac{{SS}_{NE}}{({k}_{N}^{2}-1)({k}_{N}-1)}; {MS}_{NE}\in \left[{MS}_{LE},{MS}_{UE}\right]$$

The neutrosophic \({F}_{N}\)-test is

$${F}_{N}=\frac{{MS}_{NTr\left(adj\right)}}{{MS}_{NE}}; {F}_{N}\in \left[{F}_{L},{F}_{U}\right]$$

The neutrosophic form of the \({F}_{N}\)-test for neutrosophic balanced lattice design can be expressed as:

$${F}_{N}={F}_{L}+{F}_{U}{I}_{{F}_{N}}; {I}_{{F}_{N}}\in \left[{I}_{{F}_{L}},{I}_{{F}_{U}}\right],$$

where \({I}_{{F}_{L}}\) and \({I}_{{F}_{U}}\) are determinate part and indeterminate part of the neutrosophic \({F}_{N}\)-test for neutrosophic balanced lattice design. This test statistic reduces to test statistic under classical statistic if \({I}_{{F}_{N}}=0\).

4 Numerical Example

In the following example, a researcher runs to study the effects of neutrosophic treatments on a particular study. The \(3\times 3\) neutrosophic balanced lattice design is used. The plan compares nine neutrosophic treatments in four neutrosophic replications. The data is given in Table 4.

Table 4 Data for the neutrosophic balanced lattice design
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The data is distributed in the balanced lattice design as shown in Table 5. Also included in this Table 6 is the calculation of the mathematical expressions necessary to construct the NANOVA Table and derive the \({F}_{N}\)-test for the proposed design.

Table 5 NANOVA Table for neutrosophic balanced lattice design
Full size table
Table 6 Schematic representation of the proposed design, showing the distribution of 9 treatments within the 12 incomplete blocks with 4 replications
Full size table

The proposed \({F}_{N}\)-test for NBLD will be conducted using the following steps:

  • Step 1: Data was generated using the R programming language.

  • Step 2: \({H}_{N0}:{\tau }_{N1}={\tau }_{N2}=\dots ={\tau }_{Nb}=0\;vs\;{H}_{N1}:{at\;least\;one\;\tau }_{Nk}\ne 0\).

  • Step 3: At the level of significance \(\alpha =0.05\), we calculate the \({p}_{N}-value=[0.348, 0.575]\).

  • Step 4: We accept the null hypothesis \({H}_{N0}\) because \({p}_{N}-value=\left[0.348, 0.575\right]>0.05\).

From the study, it is concluded that the nine treatments have the same mean. To keep this section concise, we will present and detail the key findings of this numerical example in the subsequent section.

5 Advantages of the Proposed Design

The aim of this section is to define the advantages of the proposed design in the presence of uncertainty. The proposed \({F}_{N}\)-test efficiency is assessed in terms of its measure of indeterminacy, adequacy, information, and flexibility. To illustrate the advantages of neutrosophic balanced lattice design, the proposed \({F}_{N}\)-test is used. The neutrosophic form of the \({F}_{N}\)-test is \({F}_{N}=1.22-0.85{I}_{N}\); \({I}_{{F}_{N}}\in [0, 0.435]\). In this neutrosophic form, there is an \(F\)-test of classical statistics and an indeterminate part. The neutrosophic form of the neutrosophic \({F}_{N}\)-test reduces to the \(F\)-test when \({I}_{{F}_{N}}=0\). This means that the value \(1.22\) represents the value of the \(F\)-test for the existing balanced lattice design under classical statistics. While, the second part \(-0.85{I}_{N}\) presents the indeterminate part with the measure of indeterminacy is \(0.435\). The information presented indicates that the proposed \({F}_{N}\)-test can take values ranging between \(1.22\) and \(0.85\). On the other hand, the existing \(F\)-test only takes into account a single value that is adequate in uncertainty. At \(\alpha\) significance level, the \({p}_{N}-value\) are \(\left[0.348, 0.575\right]>0.05\). Accordingly, the neutrosophic null hypothesis is accepted and the neutrosophic alternative hypothesis is rejected, which indicates that in the numerical example, there is no significant difference between the means of the assumed treatments. Additionally, 0.435 is the measure of indeterminacy associated with the proposed \({F}_{N}\)-test. The existing \(F\)-tests cannot provide information regarding measure of indeterminacy. In light of the results of the study, it can be concluded that the proposed \({F}_{N}\)-test is more informative and flexible than the existing \(F\)-test.

6 Future Research Directions

The literatures concerning connecting the experimental design analysis to Artificial Intelligence (AI) are few. [33, 34] handled the classical approach for machine learning settings. On the other hand, the Neutrosophic data analytics under AI was applied but not in experimental design framework [35]. As a result, we plan to extend the proposed method in this paper to encompass the AI studies in future works. We emphasize using AI because it can reduce the number of repetitions needed to attain the best experimental parameters, leading to save valuable time and resources.

7 Conclusions

This paper introduces a balanced lattice design under neutrosophic for analyzing indeterminate, uncertain, and imprecise data. The proposed design was a generalization of the existing balanced lattice design of uncertain events under classical statistics. We examined the performance of the proposed \({F}_{N}\)-test for neutrosophic balanced lattice design compared to the existing balanced lattice design test. An application of the proposed balanced lattice design was presented using numerical data. This study indicates that the proposed \({F}_{N}\)-test offers greater flexibility, applicability, and information than the existing \(F\)-test. The researchers can apply the proposed \({F}_{N}\)-test to test whether the neutrosophic treatments have the same mean or not. Research on partially balanced lattice designs under neutrosophic statistics for indeterminate data can be undertaken in the future.