# An Empowerment Scale Analysis of Mexican MSMEs: Modeling with Covariance Structures

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Empowerment

#### 2.1. Economic and Management Theories Related to Empowerment

#### 2.2. Approaches to Empowerment

#### 2.3. Theoretical Construction of Research Variables and Hypotheses

- Meaning. The collaborator connects his values, beliefs, and behaviors with the goals and objectives of the company.
- Competition. The confidence the collaborator has in his abilities, skills, and knowledge to carry out his work independently of other collaborators.
- Self-determination. The autonomy and initiative that the collaborator has to choose and make decisions to execute and develop his activities.
- Impact. The collaborator’s perception of the consequences of his actions and his influence on the results of the company.

**Hypothesis**

**H1.**

**Hypothesis**

**H2.**

**Hypothesis**

**H3.**

**Hypothesis**

**H4.**

## 3. Methods

#### 3.1. Sample and Data Collection

#### 3.2. Measurement Instruments

#### 3.3. Statistical Analysis

_{i}) and the scores of the individuals in item i (x

_{i}). They can be expressed as

- m = number of items on the scale.
- x
_{i}= items on the scale.

- m = number of items on the scale.
- x
_{i}= items on the scale.

^{2}test (high and significant value less than 0.05), Kaiser’s Measurement Adequacy Test-Meyer–Oklin (KMO), measure of sampling adequacy (MSA index), unacceptable for values below 0.5. To apply exploratory factor analysis of principal components, an analysis of principal axes with varimax rotation is performed [99]. Those indicators whose factorial load is very low (>0.5) should be suppressed [98]. This analysis allows identification of the underlying dimensions in the empowerment construct.

^{2}parameter must take values above 0.4 [101].

^{2}) and significance level (p). Both variables are very sensitive to the sample size, and when it is very high, the test may not be reliable, especially for samples greater than 200 [102,103]. Thus, the goodness of fit index (GFI) is also used, as well as the adjusted goodness of fit index (AGFI) and the Tucker Lewis index (TLI; acceptable fit for values close to 0.9; [100]), and root mean square error of approximation (RMSEA) in which, according to Maydeu-Olivares et al. [104], values around 0.08 represent a reasonable error, and Reeve et al. [105] indicate that a value of 0.05 shows an excellent fit. For normalized χ

^{2}(χ

^{2}/df), values between 2, 3, or 5 are recommended [98,100].

- −
- Chi-squared (χ
^{2}) following the maximum likelihood estimation method:$${\chi}_{ML}^{2}=\left(N-1\right)\left[Tr\left(S{\sum}^{-1}\right)-\left(p+q\right)\text{}\mathrm{ln}|\sum |-\mathrm{ln}\left|\mathrm{S}\right|\right]$$The setting function is$${F}_{ML}=\left[Tr\left(S{\sum}^{-1}\right)-\left(p+q\right)\text{}\mathrm{ln}|\sum |-\mathrm{ln}\left|\mathrm{S}\right|\right]$$- Tr = trace.
- Ln = natural logarithm.
- P = total number of estimated independent parameters.
- N = sample size.
- p + q = number of observable variables analyzed.
- df = degrees of freedom = 0.5 (p + q)(p + q + 1) − P.
- S = observed covariance matrix (initial).
- χ
^{2}= Chi squared. - ∑ = matrix of covariances reproduced by the model.

- −
- Goodness of fit index (GFI)$$GFI=1-\frac{Tr{({\sum}^{-1}S-I)}^{2}}{Tr{({\sum}^{-1}S)}^{2}}$$
- Tr = trace.
- S = observed covariance matrix (initial).
- I = Identity matrix.

- −
- Comparative fit index (CFI).$$CFI=\frac{Max\left[\left({\chi}_{Proposed\text{}Model}^{2}-d{f}_{Proposed\text{}Model}\right),\text{}0\right]}{Max\left[\left({\chi}_{Null\text{}Model}^{2}-d{f}_{Null\text{}Model}\right),\left({\chi}_{Proposed\text{}Model}^{2}-d{f}_{Proposed\text{}Model}\right),\text{}0\right]}$$
- χ
^{2}= Chi squared. - df = degrees of freedom = 0.5 (p + q) (p + q + 1) − P.

- −
- Root mean square error of approximation (RMSEA)$$RMSEA={\left[\frac{Max\text{}\left\{{\chi}^{2}-\left(\raisebox{1ex}{$df$}\!\left/ \!\raisebox{-1ex}{$N-1$}\right.\right),\text{}0\right\}}{df}\right]}^{-1/2}$$
- χ
^{2}= Chi squared. - df = degrees of freedom = 0.5 (p + q) (p + q + 1) − P.
- N = sample size.

- −
- Normed chi-square (NCS).$$NCS=\frac{{\chi}^{2}}{df}$$
- χ
^{2}= Chi squared. - df = degrees of freedom = 0.5 (p + q) (p + q + 1) − P.

- −
- Composite reliability (CR).$$CR=\frac{{({{\displaystyle \sum}}_{i=1}^{p}{\lambda}_{i})}^{2}}{{\left({{\displaystyle \sum}}_{i=1}^{p}{\lambda}_{i}\right)}^{2}+\left({{\displaystyle \sum}}_{i=1}^{p}{\epsilon}_{i}\right)}$$

- −
- Average variance extracted (AVE).$$AVE=\frac{{{\displaystyle \sum}}_{i=1}^{p}{\lambda}_{i}{}^{2}}{{{\displaystyle \sum}}_{i=1}^{p}{\lambda}_{i}{}^{2}+{{\displaystyle \sum}}_{i=1}^{p}{\epsilon}_{i}}$$

#### 3.4. Formal Establishment of the Model Structure in Confirmatory Factor Analysis

#### 3.4.1. Specification of the Measurement Model

_{x}ξ + δ.

_{x}(lambda x) is a matrix q × n of regression coefficients of X with respect to the latent variable ξ (ksi), and δ (delta) is a vector q × 1 of measurement errors of X. Factorial equations of the model are X

_{1}= λ

_{1}ξ

_{1}+ δ

_{1}… X

_{q}= λ

_{q}ξ

_{n}+ δ

_{q}.

_{y}η + ε [106] (p. 18).

_{y}(lambda y) is a p × m matrix of regression coefficients of the latent variable η (eta), and ε (epsilon) represents a vector p × 1 of measurement errors with respect to Y. Factorial equations of the model: Y

_{1}= λ

_{1}η

_{1}+ ε

_{1}… Y

_{p}= λ

_{p}η

_{m}+ ε

_{p}.

#### 3.4.2. Structural Model Specification

_{ε}are diagonals, then Ψ = ζ and Θ

_{ε}= ε, so the errors would be specified by ζ (structural model) and ε (measurement model). The vector ζ (not present in the first-order confirmatory model) provides the amount of variance of the first-order factors that is not explained by the second-order factors. The starting assumptions in the second-order models are the expected value of the measurement errors is equal to 0 (E (ζ) = 0 and E (ε

_{j}) = 0), and the expected covariation with the rest of the latent variables (ξ, η, ζ, ε) is 0 [106] (p.242).

_{yy}(θ) will contain the covariances defined, based on the unknown parameters of the model, which are found in the vector θ [106] (p. 242).

_{yy}(θ) = E

_{yy}= E(YY´) = (Λ

_{y}η + ε) (η´Λ

_{y}´ + ε´) = Λ

_{y}E(ηη´)Λ

_{y}´+ Θ

_{ε}

_{y}(ΓE(ξξ´)Γ´ + Ψ)Λ

_{y}´ + Θ

_{ε}

E(YY´) = Λ

_{y}(ΓΦΓ´ + Ψ) Λ

_{y}´ + Θ

_{ε}

## 4. Results

#### 4.1. Participant Characteristics

#### 4.2. Validation of the Scale: Exploratory Factor Analysis

#### 4.3. Second-Order Model: Confirmatory Factor Analysis

^{2}= 94.148, p = 0.001) is significantly lower than that of Model 4 (re-specified first-order model). This indicates that the data are explained more concisely by the second-order model, and better represent the internal structure of the scale. The higher the goodness of fit indices of the model collected, in Table 3, confirm that the optimal measurement model is a second-order model (Model 5); all the indices present values within the generally accepted limits (GFI = 0.941, AGFI = 0.905, CFI = 0.973, RMSEA = 0.054 χ

^{2}Normalized (χ

^{2}/df) = 1.652). Therefore, the empowerment scale is made up of two dimensions. In this model, the indicators show β > 0.50 and significant (critical coefficient> ±1.96).

## 5. Discussion

_{1}is rejected.

_{2}is partially accepted.

_{3}is partially accepted.

_{4}is rejected.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

E1 | Importance of the work done by workers |

E2 | Personal meaning of the activities carried out by the worker |

E3 | Meaning of worker performance |

E4 | Confidence in the collaborator’s ability |

E5 | Confidence in the knowledge and skills of employees |

E6 | Confidence that the employee has mastered their skills to carry out their activities |

E7 | Confidence in the worker’s autonomy over their activities |

E8 | Freedom granted to the collaborator to plan their activities |

E9 | Freedom and independence granted to the collaborator to carry out their activities |

E10 | Impact of the workers’ initiative |

E11 | Control of collaborators over their work area |

E12 | Collaborator influence on company results |

E13 | Importance given to the worker’s educational level to grant authority |

E14 | Training programs implemented in the company |

E15 | Access to the information available to collaborators |

E16 | Efficiency with which the company distributes information to employees |

E17 | Expectation of remuneration for the work initiative |

E18 | Prize and reward information available to collaborators |

E19 | Information, about clients, available to collaborators |

Note: Items that were removed appear in italics. |

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Matrix | Parameter | Dimensions | Covariances | Dimensions | Description |
---|---|---|---|---|---|

ξ | ξ (ksi) | n × 1 | Φ = E (ξξ´) | n × n | Exogenous latent variables |

X | x | q × 1 | ∑ = E ($XX$´) | q × q | Exogenous observable variables |

Λx Λy | λ (lambda) | q × n p × m | -- | -- | Factor loads of X on ξ Factor loads of Y on η |

δ | δ (delta) | q × 1 | Θδ = E (δδ´) | q × q | X measurement errors |

η | η (eta) | m × 1 | Cov(η) = E (ηη´) | m × m | Endogenous latent variables |

ζ | Ζ (zeta) | m × 1 | Ψ = E (ζζ´) | m × m | Measurement errors of η |

B | Β (beta) | m × m | -- | -- | Direct effects of η on η |

Γ | Γ (gamma) | m × n | -- | -- | Direct effects of ξ on η |

Y | Y | p × 1 | E (YY´) | p × p | Endogenous observable variables |

ε | Ε (epsilon) | p × 1 | Θε = E (εε´) | p × p | Measurement errors of Y |

Constructs | Scale Items ^{1} | Correlation Item-Total | Mean | Standard Deviation | Exploratory Factor Analysis (Loadings) | |
---|---|---|---|---|---|---|

Factor 1 | Factor 2 | |||||

Empowerment (α Cronbach: 0.870) | EM1 | 0.482 | 5.60 | 0.875 | 0.821 0.847 0.839 0.732 0.687 0.614 | 0.794 0.818 0.758 0.595 0.709 0.532 -- -- 0.475 |

EM2 | 0.510 | 5.44 | 0.943 | |||

EM3 | 0.550 | 5.59 | 0.801 | |||

EM4 | 0.629 | 5.47 | 0.822 | |||

EM5 | 0.612 | 5.41 | 0.795 | |||

EM6 | 0.611 | 5.41 | 0.795 | |||

EM7 | 0.639 | 4.81 | 1.111 | |||

EM8 | 0.538 | 4.64 | 1.209 | |||

EM9 | 0.483 | 4.77 | 1.165 | |||

EM10 | 0.548 | 5.08 | 1.003 | |||

EM11 | 0.32 | 4.81 | 1.008 | |||

EM12 | 0.543 | 5.21 | 1.021 | |||

EM13 | Eliminated | 4.42 | 1.537 | |||

EM14 | Eliminated | 4.31 | 1.512 | |||

EM15 | 0.479 | 5.15 | 1.029 | |||

EM16 | Eliminated | 4.01 | 1.549 | |||

EM17 | Eliminated | 4.69 | 1.338 | |||

EM18 | Eliminated | 4.20 | 1.738 | |||

EM19 | Eliminated | 4.33 | 1.671 | |||

Own value | 3.820 | 3.603 | ||||

% factor explained variance | 29.384 | 27.718 | ||||

% cumulative explained variance | 29.384 | 57.102 | ||||

Bartlett’s test of Sphericity Kaiser–Meyer–Oklin index ^{2} | χ^{2}(sig.): 1396.883 (0.000); KMO: 0.844; Measure of simple adequacy: (0.897–0.918);% variance: 57.102 |

^{1}The names of the items listed in this table have been shortened in order to simplify the results.

^{2}Tests that show the data obtained through the questionnaire are adequate to perform the factor analysis (requirements: Bartlett’s Sphericity Test, χ

^{2}(sig. > 0.05), Kaiser’s Measurement Adequacy Test-Meyer–Oklin (KMO) > 0.8bueno, MSA index = unacceptable for values below 0.5). Source: Authors’ own data.

Models | χ^{2} | df | χ^{2}(df) | P | GFI | AGFI | TLI | CFI | RMSEA | PRATIO |
---|---|---|---|---|---|---|---|---|---|---|

Model 1—(1 variable—13 items) | 590.438 | 65 | 9.084 | 0.000 | 0.611 | 0.508 | 0.534 | 0.611 | 0.191 | 0.833 |

Model 2—1st order (2 oblique factors—13 items) | 331.406 | 64 | 5.178 | 0.000 | 0.792 | 0.704 | 0.759 | 0.802 | 0.137 | 0.821 |

Model 3—1st order (2 orthogonal factors—13 items) | 368.100 | 65 | 5.663 | 0.000 | 0.784 | 0.698 | 0.731 | 0.776 | 0.145 | 0.833 |

Model 4—1st order (Model 2 re-specified- 2 variables—13 items) | 100.714 | 58 | 1.736 | 0.000 | 0.937 | 0.902 | 0.958 | 0.968 | 0.058 | 0.744 |

Model 5—2nd order (3 variables—13 items) | 94.148 | 57 | 1652 | 0.001 | 0.941 | 0.905 | 0.962 | 0.973 | 0.054 | 0.731 |

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**MDPI and ACS Style**

del Río-Rama, M.d.l.C.; Ríos-Manríquez, M.; Álvarez-García, J.; Sánchez-Fernández, M.D.
An Empowerment Scale Analysis of Mexican MSMEs: Modeling with Covariance Structures. *Mathematics* **2020**, *8*, 1817.
https://doi.org/10.3390/math8101817

**AMA Style**

del Río-Rama MdlC, Ríos-Manríquez M, Álvarez-García J, Sánchez-Fernández MD.
An Empowerment Scale Analysis of Mexican MSMEs: Modeling with Covariance Structures. *Mathematics*. 2020; 8(10):1817.
https://doi.org/10.3390/math8101817

**Chicago/Turabian Style**

del Río-Rama, María de la Cruz, Martha Ríos-Manríquez, José Álvarez-García, and María Dolores Sánchez-Fernández.
2020. "An Empowerment Scale Analysis of Mexican MSMEs: Modeling with Covariance Structures" *Mathematics* 8, no. 10: 1817.
https://doi.org/10.3390/math8101817