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Neutrosophic statistical analysis of split-plot designs

  • Mathematical methods in data science
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Abstract

The classic split-plot designs are unable to analyze indeterminate and uncertain data resulting from circumstances beyond our control. To this end, proposing a generalized approach to be applied to split-plot designs in uncertain environments is desired. In this study, a new approach is proposed using neutrosophic statistics to analyze split-plot and split-block designs. By such an approach neutrosophic hypothesis is formulated, a decision rule is suggested, and neutrosophic ANOVA Tables, including the FN-test, are derived. Furthermore, a numerical example and a simulation study are established to evaluate the effectiveness of the proposed designs. The results confirm that the neutrosophic logic of the proposed designs is more efficient and flexible than the classic designs in the event of facing uncertain data.

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Data availability

Enquiries about data availability should be directed to the authors.

Abbreviations

ANOVA:

Analysis of variance

MC:

Monte Carlo

NSPD:

Neutrosophic split-plot design

NSBD:

Neutrosophic split-block design

NMS:

Neutrosophic mean square

NND:

Neutrosophic normal distribution

NRV:

Neutrosophic random variable

NSS:

Neutrosophic sum of square

\({X}_{N}\) :

The neutrosophic random variable

\({n}_{N}\) :

The neutrosophic random sample selected from a population

\({\mu }_{N}\) :

The neutrosophic population mean

\({\sigma }_{N}^{2}\) :

The neutrosophic population variance

\({I}_{N}\) :

The indeterminacy interval

\({\overline{X} }_{N}\) :

The neutrosophic sample mean

\({s}_{N}^{2}\) :

The neutrosophic sample variance

\({y}_{Nhijk}\) :

The neutrosophic response of the \(Nhij{\rm th}\)split-plot experimental unit

\({\rho }_{Nh}\) :

The neutrosophic effect of the \(h\)th block or replicate

\({\tau }_{Ni}\) :

The neutrosophic effect of the \(i{\rm th}\)whole-plot

\({\eta }_{Nhi}\) :

The neutrosophic whole-plot error

\({\beta }_{Nj}\) :

The neutrosophic effect of the \(j{\rm th}\) split-plot

\({\delta }_{Nhj}\) :

The neutrosophic split-plot error

\({\left(\tau \beta \right)}_{Nij}\) :

The neutrosophic interaction effect of the \(i{\rm th}\) whole plot with the \(j{\rm th}\) split-plot

\({\varepsilon }_{Nhij}\) :

The neutrosophic interaction error or neutrosophic split-plot error

\({SS}_{NR}\) :

The neutrosophic replicate or block sum of squares

\({SS}_{NA}\) :

The neutrosophic whole-plot sum of squares

\({SS}_{NE(A)}\) :

The neutrosophic error sum of squares for whole-plot

\({SS}_{NB}\) :

The neutrosophic split-plot sum of squares

\({SS}_{NE(B)}\) :

The neutrosophic error sum of squares for split-plot

\({SS}_{NAB}\) :

The neutrosophic interaction sum of squares

\({SS}_{NE(AB)}\) :

The neutrosophic error sum of squares for interaction

\({SS}_{NT}\) :

The neutrosophic total sum of squares

\({MS}_{NR}\) :

The neutrosophic replicate or block mean squares

\({MS}_{NA}\) :

The neutrosophic whole-plot mean squares

\({MS}_{NE(A)}\) :

The neutrosophic error mean squares for whole-plot

\({MS}_{NB}\) :

The neutrosophic split-plot mean squares

\({MS}_{NE(B)}\) :

The neutrosophic error mean squares for split-plot

\({MS}_{NAB}\) :

The neutrosophic interaction mean squares

\({MS}_{NE(AB)}\) :

The neutrosophic error mean squares for interaction

\({F}_{NA}\) :

The neutrosophic whole-plot \(f-\) distribution

\({F}_{NB}\) :

The neutrosophic split-plot \(f-\) distribution

\({F}_{NAB}\) :

The neutrosophic interaction \(f-\) distribution

\({p}_{N}-value\) :

The neutrosophic p value

\(\alpha \) :

Level of significance

\({F}_{N}\) :

The neutrosophic \(f-\) distribution

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Acknowledgements

The authors are deeply thankful to the editor and reviewers for their valuable suggestions to improve the quality and presentation of the paper.

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Authors and Affiliations

  1. Department of Statistics, University of Isfahan, Isfahan, Iran

    Abdulrahman AlAita & Hooshang Talebi

  2. Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

    Muhammad Aslam

  3. Department of Agricultural Economy, Faculty of Agriculture, Damascus University, Damascus, Syria

    Khaled Al Sultan

Authors
  1. Abdulrahman AlAita
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  2. Hooshang Talebi
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  3. Muhammad Aslam
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  4. Khaled Al Sultan
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Correspondence to Muhammad Aslam.

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AlAita, A., Talebi, H., Aslam, M. et al. Neutrosophic statistical analysis of split-plot designs. Soft Comput 27, 7801–7811 (2023). https://doi.org/10.1007/s00500-023-08025-y

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