inst/help/PrincipalComponentAnalysis.md

In jasp-stats/jaspFactor: Factor Module for JASP

Principal Component Analysis

Principcal Component Analysis is used to represent the data in smaller components than the dataset originally consists of. The components are chosen such that they explain most of the variance in the original dataset.

Assumptions

• The variables included in the analysis are correlated (Shlens, 2014).
• The variables included in the analysis are linearly related (Shlens, 2014).

Input

Assignment Box

• Included Variables: In this box, the variables to perform the principal component analysis on are selected.

Number of Components

• Here, the number of components that the rotation is applied to is specified. Several methods to determine this number can be chosen from:
• Parallel Analysis: Factors are selected on the basis of parallel analysis. With this method, factors are selected when their eigenvalue is greater than the parallel average random eigenvalue. This method is selected by default. Can be based on principal component eigenvalues (PC) or factor eigenvalues (FA). A seed (1234) is chosen by default so that the results from the parallel analysis are equal across the PCA.
• Eigenvalues: Components are selected when they have a certain eigenvalue. By default components are selected that have an eigenvalue above 1.
• Manual: The number of components can be specified manually. By default this is set to 1.

Rotation

• Here, the rotation method to apply to the components can be specified. Rotation ensures a simpler understanding of the data structure.
• Orthogonal: This method produces components that are uncorrelated. For this method, there are several possibilities that can be selected:
  • None: No rotation method is selected.
  • varimax: Orthogonal rotation method varimax. Rotation based on the maximizing the variance of the loadings.
  • quartimax: Orthogonal rotation method quartimax. In this rotation method, the number of components that is necessary to explain each variable is minimized.
  • bentlerT: Orthogonal rotation method bentlerT.
  • equamax: Orthogonal rotation method equamax. This is a combination of varimax and quartimax.
  • varimin: Orthogonal rotation method varimin.
• Oblique: This method produces components that allow for correlation between the components. This method is selected by default. Several possibilities are available:
  • promax: Oblique rotation method promax. This method is selected by default.
  • oblimin: Oblique rotation method oblimin.
  • simplimax: Oblique rotation method simplimax.
  • bentlerQ: Oblique rotation method bentlerQ.
  • biquartimin: Oblique rotation method biquartimin.
  • cluster: Oblique rotation method cluster.

Base decomposition on

• Correlation: Bases the PCA on the correlation matrix of the data
• Covariance: Bases the PCA on the covariance matrix of the data
• Polychoric/tetrachoric: Bases the PCA on the poly/tetrachoric (mixed) correlation matrix of the data. This is sometimes unstable when sample size is small and when some variables do not contain all response categories

Output Options

• Highlight: This option cuts the scaling of paths in width and color saturation. Paths with absolute weights over this value will have the strongest color intensity and become wider the stronger they are, and paths with absolute weights under this value will have the smallest width and become vaguer the weaker the weight. If set to 0, no cutoff is used and all paths vary in width and color.
• Include Tables:
  • Component correlations: When selecting this option, a table with the correlations between the components will be displayed.
  • Residual matrix: Displays a table containing the residual variances and correlations
  • Parallel analysis: If this option is selected, a table will be generated exhibiting a detailed output of the parallel analysis. Can be based on principal component eigenvalues (PC) or factor eigenvalues (FA). The seed is taken from the parallel analysis for determining the number of factors above.
  • Path diagram: By selecting this option, a visual representation of the direction and strength of the relation between the variable and component will be displayed.
  • Scree plot: When selecting this option, a scree plot will be displayed. The scree plot provides information on how much variance in the data, indicated by the eigenvalue, is explained by each component. A scree plot can be used to decide how many components should be selected.
• Assumption Checks:
• Kaiser-Meyer-Olkin Test (KMO): Determines how well variables are suited for factor analysis by computing the proportion of common variance between variables
• Bartlett's Test (of sphericity): Determines if the data correlation matrix is the identity matrix, meaning, if the variables are related or not
  • Mardia's Test of Multivariate Normality: Assesses the degree of the departure from multivariate normality of the included variables in terms of multivariate skewness and kurtosis. The Mardia's test will always include the listwise complete cases.
• Missing values:
  • Exclude cases pairwise: If one observation from a variable is missing, all the other variable observations from the same case will still be used for the analysis. In this scenario, it is not necessary to have a observation for all the variables to include the case in the analysis. This option is selected by default.
  • Exclude cases listwise: If one observation from a variable is missing, the whole case, so all the other connected variable observations, will be dismissed from the analysis. In this scenario, observations for every variable are needed to include the case in the analysis.

Output

Assumption Checks

• Kaiser-Meyer-Olkin Test (KMO): Measure of sampling adequacy (MSA) as the proportion of common variance among variables is computed for all variables; values closer to 1 are desired.
• Bartlett's Test (of sphericity): A significant result means the correlation matrix is unlike the identity matrix.
• Mardia's Test of Multivariate Normality:
  • Tests: The first column shows all the tests performed.
  • Value: The values of b1p (multivariate skewness) and b2p (multivariate kurtosis), as denoted in Mardia (1970).
  • Statistic: The two chi-squared test statistics of multivariate skewness (both standard and corrected for small samples) and the standard normal test statistic of multivariate kurtosis.
  • df: Degrees of freedom.
  • p: P-value.

Chi-squared Test:

The fit of the model is tested. When the test is significant, the model is rejected. Bear in mind that a chi-squared approximation may be unreliable for small sample sizes, and the chi-squared test may too readily reject the model with very large sample sizes. See, for example, Saris, Satorra, & van der Veld (2009) for more discussions on overall fit metrics. - Model: The model obtained from the principal component analysis. - Value: The chi-squared test statistic. - df: Degrees of freedom. - p: P-value.

Component Loadings:

• Variables: The first column shows all the variables included in the analysis.
• PC (1, 2, 3, ...): This column shows the variable loadings on the components.
• Uniqueness: The percentage of the variance of each variable that is not explained by the component.

Component Characteristics:

• Unrotated solution:
• Eigenvalues: The eigenvalue for each component
• Proportion var.: The proportion of variance in the dataset explained by each unrotated component
• Cumulative: The proportion of variance in the dataset explained by the unrotated components up to and including the current component.
• Rotated solution:
• SumSq. Loadings: Sum of squared loadings, variance explained by each rotated component
• Proportion var.: The proportion of variance in the dataset explained by each rotated component

• Cumulative: The proportion of variance in the dataset explained by the rotated components up to and including the current component.

• Correlations: The correlation between the principal components.

Path Diagram

• PC: The principal components are represented by the circles.
• Variables: The variables loadings on the components are represented by the boxes.
• Arrows: Going from the variables to the principal components, representing the loading from the variable on the component. Red indicates a negative loading, green a positive loading. The wider the arrows, the higher the loading. This highlight can be adjusted at highlight in the Output Options.

Screeplot

The scree plot provides information on how much variance in the data, indicated by the eigenvalue, is explained by each component. The scree plot can be used to decide how many components should be selected in the model. - Components: On the x-axis, the components. - Eigenvalue: On the y-axis, the eigenvalue that indicates the variance explained by each component. - Data: The dotted line represents the data. - Simulated: The triangle line represents the simulated data. This line is indicative for the parallel analysis. When the points from the dotted line (real data) are above this line, these components will be included in the model by parallel analysis. - Kaiser criterion: The horizontal line at the eigenvalue of 1 represents the Kaiser criterion. According to this criterion, only components with values above this line (at an eigenvalue of 1) should be included in the model.

References

• Dziuban, C. D., & Shirkey, E. C. (1974). When is a correlation matrix appropriate for factor analysis? Some decision rules. Psychological Bulletin, 81(6), 358–361. https://doi.org/10.1037/h0036316
• Hayton, J. C., Allen, D. G., & Scarpello, V. (2004). Factor retention decisions in exploratory factor analysis: A tutorial on parallel analysis. Organizational Research Methods, 7(2), 191-205. https://doi.org/10.1177/1094428104263675
• Horn, J. L. (1965). A rationale and test for the number of factors in factor analysis. Psychometrika, 30(2), 179–185. https://doi.org/10.1007%2Fbf02289447
• James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An introduction to statistical learning (2nd ed.). Springer.
• Mardia, K. V. (1970). Measures of multivariate skewness and kurtosis with applications. Biometrika, 57(3), 519-530. https://doi.org/10.2307/2334770
• Osborne, J. W., Costello, A. B., & Kellow, J. T. (2008). Best practices in exploratory factor analysis. In J. Osborne (Ed.), Best practices in quantitative methods (pp. 86-99). SAGE Publications, Inc. https://doi.org/10.4135/9781412995627.d8
• Saris, W. E., Satorra, A., & Van der Veld, W. M. (2009). Testing structural equation models or detection of misspecifications?. Structural Equation Modeling: A Multidisciplinary Journal, 16(4), 561-582. https://doi.org/10.1080/10705510903203433
• Shlens, J. (2014). A tutorial on principal component analysis. arXiv preprint arXiv:1404.1100. https://doi.org/10.48550/arXiv.1404.1100

R Packages

• psych
• qgraph

jasp-stats/jaspFactor documentation built on April 20, 2024, 4:12 p.m.