# PLS visualization using biplots: an application to team effectiveness

## Summary (4 min read)

### 1 Introduction

- Frequently, multivariate data analysis seeks to perceive the existing underlying structure and to understand the relationships established within data.
- Likewise, a graph of the results of a specific statistical method, e.g., the Principal Component Analysis (PCA) biplot, enhances data familiarity.
- This is the case of an ill-posed problem, in which the predictors are many and quasi-collinear, leading to an unstable Ordinary Least Squares (OLS) solution, i.e., the OLS estimates have high variance (Belsley, Kuh & Welsch, 2004).
- Based on the outputs of the PLS (scores, loadings, and weights vectors), the variances and correlations of the variables can be revealed by employing an exploratory PLS biplot.
- The primary purpose of this article is to provide a straightforward interpretation for the PLS biplot applicable to both exploratory and predictive purposes, illustrating its application in team effectiveness research data.

### 2.1 Partial Least Squares

- One might use the PLS to estimate the regression coefficients.
- The PLS model consists of three other models, two external and one internal, as a result of the application of a suitable algorithm, usually the Nonlinear Iterative Partial Least Squares .
- The method seeks to estimate some underlying factors that decompose X and Y simultaneously, maximizing the covariance between them, establishing the socalled outer relations for X and Y individually (Geladi & Kowalsky, 1986).
- Considering the extraction of all possible factors, the PLS decomposition results in X = TP' and Y = UQ', where T contains the scores of the predictors’ matrix, P holds the loadings of X.

### 2.2 Partial Least Squares Regression

- Concisely, the NIPALS algorithm1 performs the following steps (Abdi, 2010): - Step 1. w ∝ X'u (X-weights).

### 2.3 The Biplot

- The term biplot was introduced by Gabriel (1971) and consists of a graphical representation that reveals important characteristics of data structure, e.g., patterns of correlations between variables or similarities between the observations (Greenacre, 2010).
- The matrices G and H that arise from the decomposition of D create two sets of points.

### 2.4 The Exploratory PLS Biplot

- The symbol ∝ means ‘to normalize the result of the operation’.
- So, the rows of T represent the biplot points in the exploratory PLS biplot, expressing the observations of the sample, while the columns of the block matrix [P Q]' indicate the biplot vectors and denote the variables, wherein those from column 1 to m refer to the predictors and from column (m + 1) to (m + q) are associated with the responses.
- Considering each set of biplot vectors separately (predictors and responses), the angle formed by two vectors provides an approximation for the sample correlation coefficient related to the associated variables (Grafelman, 2012).
- Therefore, if ∠(𝐩i ′, 𝐩j ′) ≅ 0°, it means that the associated variables are strongly correlated because the cosine of the angle between the biplot vectors is close to one.
- The accuracy of this approximation will depend on how much the variables contribute to each of the underlying components estimated (Bassani, Ambrogi, Coradini, & Biganzoli, 2010), as well as the biplot explained variance (Greenacre, 2012).

### 2.5 The Predictive PLS Biplot

- In the predictive PLS biplot, the rows of the matrix R denote the biplot points instead of the rows of T. Further, the columns of Q' symbolize the responses through biplot vectors.
- Each response can also define a calibrated axis, on which one can project the set of points (𝐫i ′) to get an approximation of the coefficients.
- Therefore, there are two ways to evaluate an approximation for these estimates in the biplot visually.
- The area and position of the triangles furnish other relevant information about the PLS regression coefficients, such as the signal and the importance of each predictor to the model.

### 3 Application

- Teams of individuals working together to achieve a common goal are a central part of daily life in modern organizations (Mathieu, Tannenbaum, Donsbach, & Alliger, 2014).
- By bringing together individuals with different skills and knowledge, teams emerge as a competitive asset in the ever-changing organizational environment.
- When teams are created, the ultimate goal is to generate value for the organization.
- Accordingly, studying team effectiveness and the conditions that enable the team to be effective has been a central concern for both research and practice (Kozlowski & Ilgen, 2006).

### 3.1 Variables

- In the present research, in line with previous studies (e.g., Hackman, 1987), the authors consider team effectiveness as a multidimensional construct.
- Team effectiveness will be considered, in the present study, as the result of the role played by thirteen variables: team trust (2 dimensions), team psychological capital (4 dimensions), collective behavior, transformational leadership, intragroup conflict (2 dimensions), team psychological safety, and team cohesion (2 dimensions).
- Transformational leadership (X8) can be defined as a leadership style that encourages followers to do more than they originally expected, broadening and changing their interests and leading to conscientiousness and acceptance of the team’s purposes (Bass, 1990).
- It is a confidence climate that comes from mutual respect and trust between members (Edmondson, 1999).
- Task cohesion (X12) refers to the shared commitment among members towards achieving a goal that requires the collective efforts of the group.

### 3.2 Sample and Data Collection Procedure

- Organizations were selected by convenience, using the personal and professional contacts network of the research team.
- Data was obtained from 104 teams and their respective leaders.
- Missing values in the questionnaires were replaced by the item average (in case of a random distribution) or by expectation-maximization (EM) method (in case of a non-random distribution).
- Of the team members (N = 353), 67% were female, 63.3% had secondary education or less, with the remaining 36.7 % having a higher education background.
- The mean age was approximately 38 years old (SD = 12.33).

### 3.3 Measures

- Apart from team performance that was assessed by team leaders, all variables were measured by team members.
- Team trust and team psycap were assessed using 6-point scales, intragroup conflict and team psychological safety were evaluated on 7-point scales and the remaining variables were measured on 5-point scales.

### 3.4 PLS Biplot Results

- In order to reveal a linear relation between the variables describing team effectiveness and the explanatory variables, the PLS was used to construct the external and internal models.
- Next, the NIPALS algorithm was used to decompose the data matrices and to extract two PLS components, yielding the matrices 𝐓82 ×2 = [T1 T2], 𝐏13 ×2, 𝐔82 ×2, 𝐐4 ×2, 𝐖13 ×2, 𝐑13 ×2, and 𝐁13 ×2.
- Table 2 shows some significantly correlated variables evidenced by the biplot (X2 and X12, X7 and X12, X3 and X11, and X9 and X10), a pair of variables that displayed negative correlation (X2 and X9), and others that manifested a weak correlation visually (X5 and X13, X6 and X13), all of them flanked by the exact sample correlation coefficients.
- For comparison purposes only, Fig. 2 shows the results of the area biplot method.
- On the negative side, the predictors Task conflict (X9) and Affective conflict (X10) are the most influential in the model, while the explanatory variables Team trust-task (X2), Task cohesion (X12), and Social cohesion (X13) have the most significant and positive impact concerning Y2.

### 4 Discussion and Conclusions

- Regarding the application of the method the authors use in this work, the results point to the “validity” of such an application concerning the relationships found between the group processes and the team output variables considered.
- These results are in line with the literature.
- Task cohesion tends to be positively associated with team outcomes, but social cohesion can have a more complex relationship with team outcomes due the fact that social cohesion, although it increases the willingness to help each other and to cooperate, can also lead to uncritical acceptance of solutions and to groupthink (Janis, 1972).
- Overall, the studies tend to suggest that, in certain circumstances, task conflict may be positively related to group outcomes (e.g., De Wit et al., 2012) emphasizing the role of moderators, such as the conflict-handling strategies used in the team.
- The authors should keep in mind that biplot is a visualization method whose purpose is to provide a general idea of latent structures in the data, not to mention that the interpretation technique suggested in this paper provides only an approximation of the coefficients, which will be closer to the real values of the estimates, the higher the PLS components’ ability to explain the variance.

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### "PLS visualization using biplots: an..." refers background in this paper

...…research suggests that when team members have a collective belief in their ability to be effective, they explore and share knowledge and are more prepared to implement new ways of achieving results, because they believe these behaviors will lead to higher levels of performance (Bandura 1977)....

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...Additionally, previous research suggests that when team members have a collective belief in their ability to be effective, they explore and share knowledge and are more prepared to implement new ways of achieving results, because they believe these behaviors will lead to higher levels of performance (Bandura 1977)....

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### "PLS visualization using biplots: an..." refers background in this paper

...Regarding the relationship between team psychological safety and team performance, previous studies suggest that team performance can be facilitated, directly or indirectly, by the presence of a psychological security climate (e.g., Edmondson 1999)....

[...]

...It is a confidence climate that comes from mutual respect and trust between members (Edmondson 1999)....

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