pca total variance spsspca total variance spss

As a data analyst, the goal of a factor analysis is to reduce the number of variables to explain and to interpret the results. (a) Principal component analysis as an exploratory tool for data analysis. These are essentially the regression weights that SPSS uses to generate the scores. These now become elements of the Total Variance Explained table. In practice, you would obtain chi-square values for multiple factor analysis runs, which we tabulate below from 1 to 8 factors. This is because unlike orthogonal rotation, this is no longer the unique contribution of Factor 1 and Factor 2. correlation). Recall that for a PCA, we assume the total variance is completely taken up by the common variance or communality, and therefore we pick 1 as our best initial guess. Summing the squared loadings of the Factor Matrix across the factors gives you the communality estimates for each item in the Extraction column of the Communalities table. From the Factor Correlation Matrix, we know that the correlation is \(0.636\), so the angle of correlation is \(cos^{-1}(0.636) = 50.5^{\circ}\), which is the angle between the two rotated axes (blue x and blue y-axis). The eigenvector times the square root of the eigenvalue gives the component loadingswhich can be interpreted as the correlation of each item with the principal component. Method 2: Suppose I wanted to include enough principal components to explain 90% of the total variability explained by all 13 principal components. There is an argument here that perhaps Item 2 can be eliminated from our survey and to consolidate the factors into one SPSS Anxiety factor. Extraction Method: Principal Axis Factoring. Promax is an oblique rotation method that begins with Varimax (orthgonal) rotation, and then uses Kappa to raise the power of the loadings. Lets say you conduct a survey and collect responses about peoples anxiety about using SPSS. Promax also runs faster than Direct Oblimin, and in our example Promax took 3 iterations while Direct Quartimin (Direct Oblimin with Delta =0) took 5 iterations. The total common variance explained is obtained by summing all Sums of Squared Loadings of the Initial column of the Total Variance Explained table. In words, this is the total (common) variance explained by the two factor solution for all eight items. Rotation Method: Oblimin with Kaiser Normalization. Rotation Method: Varimax without Kaiser Normalization. True or False, When you decrease delta, the pattern and structure matrix will become closer to each other. In oblique rotation, an element of a factor pattern matrix is the unique contribution of the factor to the item whereas an element in the factor structure matrix is the. Just as in PCA, squaring each loading and summing down the items (rows) gives the total variance explained by each factor. A company director wanted to hire another employee for his company and was looking for someone who would display high levels of motivation, dependability, enthusiasm and commitment (i.e., these are the four constructs we are interested in). Answers: 1. I am trying to understand Principal Component Analysis (PCA). The Factor Transformation Matrix tells us how the Factor Matrix was rotated. F, greater than 0.05, 6. In fact, SPSS caps the delta value at 0.8 (the cap for negative values is -9999). With this code, I'm able to reproduce the SPSS Principal Component "Factor Analysis" result using this dataset. In the section, Procedure, we illustrate the SPSS Statistics procedure that you can use to carry out PCA on your data. 16 This question already has answers here : PCA and proportion of variance explained (4 answers) Closed 8 years ago. You will notice that these values are much lower. Unbiased scores means that with repeated sampling of the factor scores, the average of the predicted scores is equal to the true factor score. a. Predictors: (Constant), I have never been good at mathematics, My friends will think Im stupid for not being able to cope with SPSS, I have little experience of computers, I dont understand statistics, Standard deviations excite me, I dream that Pearson is attacking me with correlation coefficients, All computers hate me. However, an annoying problem are the missing percent signs. The computations are based on a correlation or covariance matrix. This number matches the first row under the Extraction column of the Total Variance Explained table. This continues until a total of p principal components have been calculated, equal to the original number of variables. For both PCA and common factor analysis, the sum of the communalities represent the total variance. We notice that each corresponding row in the Extraction column is lower than the Initial column. Without rotation, the first factor is the most general factor onto which most items load and explains the largest amount of variance. Under Total Variance Explained, we see that the Initial Eigenvalues no longer equals the Extraction Sums of Squared Loadings. Decrease the delta values so that the correlation between factors approaches zero. Factor rotations help us interpret factor loadings. The Rotated Factor Matrix table tells us what the factor loadings look like after rotation (in this case Varimax). In our case, Factor 1 and Factor 2 are pretty highly correlated, which is why there is such a big difference between the factor pattern and factor structure matrices. F, the total Sums of Squared Loadings represents only the total common variance excluding unique variance, 7. 2 factors extracted. There are two general types of rotations, orthogonal and oblique. &+ (0.036)(-0.749) +(0.095)(-0.2025) + (0.814) (0.069) + (0.028)(-1.42) \\ Remember to interpret each loading as the zero-order correlation of the item on the factor (not controlling for the other factor). Additionally, NS means no solution and N/A means not applicable. Summing the squared loadings of the Factor Matrix down the items gives you the Sums of Squared Loadings (PAF) or eigenvalue (PCA) for each factor across all items. For the second factor FAC2_1 (the number is slightly different due to rounding error): $$ In SPSS, there are three methods to factor score generation, Regression, Bartlett, and Anderson-Rubin. You can learn more about our enhanced content on our Features: Overview page. &(0.284) (-0.452) + (-0.048)(-0.733) + (-0.171)(1.32) + (0.274)(-0.829) \\ Varimax rotation is the most popular orthogonal rotation. It looks like here that the p-value becomes non-significant at a 3 factor solution. For orthogonal rotations, use Bartlett if you want unbiased scores, use the Regression method if you want to maximize validity and use Anderson-Rubin if you want the factor scores themselves to be uncorrelated with other factor scores. Computation Given a data matrix with p variables and n samples, the data are rst centered on the means of each variable. The benefit of doing an orthogonal rotation is that loadings are simple correlations of items with factors, and standardized solutions can estimate the unique contribution of each factor. Although SPSS Anxiety explain some of this variance, there may be systematic factors such as technophobia and non-systemic factors that cant be explained by either SPSS anxiety or technophbia, such as getting a speeding ticket right before coming to the survey center (error of meaurement). As an exercise, lets manually calculate the first communality from the Component Matrix. This means even if you use an orthogonal rotation like Varimax, you can still have correlated factor scores. Suppose you are conducting a survey and you want to know whether the items in the survey have similar patterns of responses, do these items hang together to create a construct? For example, Item 1 is correlated \(0.659\) with the first component, \(0.136\) with the second component and \(-0.398\) with the third, and so on. We will begin with variance partitioning and explain how it determines the use of a PCA or EFA model. This means that the sum of squared loadings across factors represents the communality estimates for each item. Each squared element of Item 1 in the Factor Matrix represents the communality. The column Extraction Sums of Squared Loadings is the same as the unrotated solution, but we have an additional column known as Rotation Sums of Squared Loadings. How do we obtain this new transformed pair of values? F, the Structure Matrix is obtained by multiplying the Pattern Matrix with the Factor Correlation Matrix, 4. Summing the squared component loadings across the components (columns) gives you the communality estimates for each item, and summing each squared loading down the items (rows) gives you the eigenvalue for each component. The angle of axis rotation is defined as the angle between the rotated and unrotated axes (blue and black axes). In this "quick start" guide, we show you how to carry out PCA using SPSS Statistics, as well as the steps you'll need to go through to interpret the results from this test. Overview This seminar will give a practical overview of both principal components analysis (PCA) and exploratory factor analysis (EFA) using SPSS. The only difference is under Fixed number of factors Factors to extract you enter 2. In order to select candidates for interview, he prepared a questionnaire consisting of 25 questions that he believed might answer whether he had the correct candidates. You can specify one of five options for normalizing the object scores and the variables. Therefore the first component explains the most variance, and the last component explains the least. &= -0.115, This can be confirmed by the Scree Plot which plots the eigenvalue (total variance explained) by the component number. The following applies to the SAQ-8 when theoretically extracting 8 components or factors for 8 items: Answers: 1. Summing the eigenvalues (PCA) or Sums of Squared Loadings (PAF) in the Total Variance Explained table gives you the total common variance explained. The Total Variance Explained table contains the same columns as the PAF solution with no rotation, but adds another set of columns called Rotation Sums of Squared Loadings. Answers: 1. This option optimizes the association between variables. Orthogonal rotation assumes that the factors are not correlated. Rotation Sums of Squared Loadings (Varimax), Rotation Sums of Squared Loadings (Quartimax). Comparing this to the table from the PCA we notice that the Initial Eigenvalues are exactly the same and includes 8 rows for each factor. For example, Component 1 is \(3.057\), or \((3.057/8)\% = 38.21\%\) of the total variance. This is not uncommon when working with real-world data rather than textbook examples. In order to generate factor scores, run the same factor analysis model but click on Factor Scores (Analyze Dimension Reduction Factor Factor Scores). Now that we understand partitioning of variance we can move on to performing our first factor analysis. For example, Factor 1 contributes \((0.653)^2=0.426=42.6\%\) of the variance in Item 1, and Factor 2 contributes \((0.333)^2=0.11=11.0%\) of the variance in Item 1. We can calculate the first component as. Additionally, since the common variance explained by both factors should be the same, the Communalities table should be the same. Notice that the contribution in variance of Factor 2 is higher \(11\%\) vs. \(1.9\%\) because in the Pattern Matrix we controlled for the effect of Factor 1, whereas in the Structure Matrix we did not. The 18 steps below show you how to analyse your data using PCA in SPSS Statistics when none of the five assumptions in the previous section, Assumptions, have been violated. You will see that whereas Varimax distributes the variances evenly across both factors, Quartimax tries to consolidate more variance into the first factor. In oblique rotations, the sum of squared loadings for each item across all factors is equal to the communality (in the SPSS Communalities table) for that item. The benefit of Varimax rotation is that it maximizes the variances of the loadings within the factors while maximizing differences between high and low loadings on a particular factor. 3. Note that in the Extraction of Sums Squared Loadings column the second factor has an eigenvalue that is less than 1 but is still retained because the Initial value is 1.067. Do all these items actually measure what we call SPSS Anxiety? Each row should contain at least one zero. Total variance explained, rotated factors. F, sum all Sums of Squared Loadings from the Extraction column of the Total Variance Explained table, 6. &= -0.880, F, delta leads to higher factor correlations, in general you dont want factors to be too highly correlated. I found a webpage on PCA that introduces it and the concept of the percentage of variance. If you do oblique rotations, its preferable to stick with the Regression method. Finally, lets conclude by interpreting the factors loadings more carefully. PCA is designed to transform the original variables into new, uncorrelated variables (axes), called the principal components, which are linear combinations of the original variables. The most common type of orthogonal rotation is Varimax rotation. Modeler's PCA/Factor Node is in effect feeding data into a module using the SPSS Statistics FACTOR procedure to do the computations. The more correlated the factors, the more difference between pattern and structure matrix and the more difficult to interpret the factor loadings. This is because rotation does not change the total common variance. These elements represent the correlation of the item with each factor. Under Extraction Method, pick Principal components and make sure to Analyze the Correlation matrix. Additionally, for Factors 2 and 3, only Items 5 through 7 have non-zero loadings or 3/8 rows have non-zero coefficients (fails Criteria 4 and 5 simultaneously). In contrast, common factor analysis assumes that the communality is a portion of the total variance, so that summing up the communalities represents the total common variance and not the total variance. If there is no unique variance then common variance takes up total variance (see figure below). This is expected because we assume that total variance can be partitioned into common and unique variance, which means the common variance explained will be lower. Note that as you increase the number of factors, the chi-square value and degrees of freedom decreases but the iterations needed and p-value increases. This makes sense because the Pattern Matrix partials out the effect of the other factor. F, the total variance for each item, 3. However, use caution when interpretation unrotated solutions, as these represent loadings where the first factor explains maximum variance (notice that most high loadings are concentrated in first factor). In the Factor Structure Matrix, we can look at the variance explained by each factor not controlling for the other factors. 1. Method 3: Here, we want to "find the elbow.". Lets compare the same two tables but for Varimax rotation: If you compare these elements to the Covariance table below, you will notice they are the same. The first ordered pair is \((0.659,0.136)\) which represents the correlation of the first item with Component 1 and Component 2. Kaiser normalization weights these items equally with the other high communality items. Eigenvalues are also the sum of squared component loadings across all items for each component, which represent the amount of variance in each item that can be explained by the principal component. $$(0.588)(0.773)+(-0.303)(-0.635)=0.455+0.192=0.647.$$. &(0.005) (-0.452) + (-0.019)(-0.733) + (-0.045)(1.32) + (0.045)(-0.829) \\ You will get eight eigenvalues for eight components, which leads us to the next table. For example, \(0.653\) is the simple correlation of Factor 1 on Item 1 and \(0.333\) is the simple correlation of Factor 2 on Item 1. The standard context for PCA as an exploratory data analysis tool involves a dataset with observations on p numerical variables, for each of n entities or individuals. Starting from the first component, each subsequent component is obtained from partialling out the previous component. Lets take a look at how the partition of variance applies to the SAQ-8 factor model. Notice that the original loadings do not move with respect to the original axis, which means you are simply re-defining the axis for the same loadings. Lets calculate this for Factor 1: $$(0.588)^2 + (-0.227)^2 + (-0.557)^2 + (0.652)^2 + (0.560)^2 + (0.498)^2 + (0.771)^2 + (0.470)^2 = 2.51$$. F, the eigenvalue is the total communality across all items for a single component, 2. The dataset for the exploratory fa. Principal components analysis (PCA, for short) is a variable-reduction technique that shares many similarities to exploratory factor analysis. Among the three methods, each has its pluses and minuses. For this particular PCA of the SAQ-8, the eigenvector associated with Item 1 on the first component is \(0.377\), and the eigenvalue of Item 1 is \(3.057\). Compare the plot above with the Factor Plot in Rotated Factor Space from SPSS. The factor pattern matrix represent partial standardized regression coefficients of each item with a particular factor. Pasting the syntax into the Syntax Editor gives us: The output we obtain from this analysis is. The coefficient matrix is p-by-p.Each column of coeff contains coefficients for one principal component, and the columns are in descending order of component variance. In oblique rotation, the factors are no longer orthogonal to each other (x and y axes are not \(90^{\circ}\) angles to each other). The total variance explained by both components is thus \(43.4\%+1.8\%=45.2\%\). You can find out about our enhanced content as a whole on our Features: Overview page, or more specifically, learn how we help with testing assumptions on our Features: Assumptions page. The second principal component is calculated in the same way, with the condition that it is uncorrelated with (i.e., perpendicular to) the first principal component and that it accounts for the next highest variance. The sum of rotations \(\theta\) and \(\phi\) is the total angle rotation. Each item has a loading corresponding to each of the 8 components. The SPSS Statistics procedure for PCA is not linear (i.e., only if you are lucky will you be able to run through the following 18 steps and accept the output as your final results). The other main difference between PCA and factor analysis lies in the goal of your analysis. It bins variance of the items. For example, to obtain the first eigenvalue we calculate: $$(0.659)^2 + (-.300)^2 + (-0.653)^2 + (0.720)^2 + (0.650)^2 + (0.572)^2 + (0.718)^2 + (0.568)^2 = 3.057$$. The goal of a PCA is to replicate the correlation matrix using a set of components that are fewer in number and linear combinations of the original set of items. However, the procedure is identical. Published with written permission from SPSS Statistics, IBM Corporation. If the total variance is 1, then the communality is \(h^2\) and the unique variance is \(1-h^2\). Lets now move on to the component matrix. Here is the output of the Total Variance Explained table juxtaposed side-by-side for Varimax versus Quartimax rotation. Take the example of Item 7 Computers are useful only for playing games. a) In principal component analysis is used (only), in order to reduce the itemset from many to few. Technically, when delta = 0, this is known as Direct Quartimin. As observed previously, the total variance for the nine random variables is 9 (since the variance was standardized to 1 in the correlation matrix), which is, as expected, equal to the sum of the nine eigenvalues listed in Figure 5. and you get back the same ordered pair. These three components explain 84.1% of the variation in the data. Looking at absolute loadings greater than 0.4, Items 1,3,4,5 and 7 loading strongly onto Factor 1 and only Item 4 (e.g., All computers hate me) loads strongly onto Factor 2. The questions were phrased such that these qualities should be represented in the questions. In addition to the option that is already selected by default (i.e.. Deutsche Bahn Regional operates a train from Eschborn-Niederhchstadt Bahnhof to Kronberg every 30 minutes. Tickets cost 3 - 5 and the journey takes 11 min. F, represent the non-unique contribution (which means the total sum of squares can be greater than the total communality), 3. Solution: Using the conventional test, although Criteria 1 and 2 are satisfied (each row has at least one zero, each column has at least three zeroes), Criterion 3 fails because for Factors 2 and 3, only 3/8 rows have 0 on one factor and non-zero on the other. Popular answers (1) Terlumun Adagba. In this case, the angle of rotation is \(cos^{-1}(0.773) =39.4 ^{\circ}\). It is usually more reasonable to assume that you have not measured your set of items perfectly. Factor analysis is based on the correlation matrix of the variables involved, and correlations usually need a large sample size before they stabilize. Here you see that SPSS Anxiety makes up the common variance for all eight items, but within each item there is specific variance and error variance. As we mentioned before, the main difference between common factor analysis and principal components is that factor analysis assumes total variance can be partitioned into common and unique variance, whereas principal components assumes common variance takes up all of total variance (i.e., no unique variance). First we bold the absolute loadings that are higher than 0.4. Eigenvalues close to zero imply there is item multicollinearity, since all the variance can be taken up by the first component. Notice that the Extraction column is smaller than the Initial column because we only extracted two components. Both methods try to reduce the dimensionality of the dataset down to fewer unobserved variables, but whereas PCA assumes that there common variances takes up all of total variance, common factor analysis assumes that total variance can be partitioned into common and unique variance. Additionally, Anderson-Rubin scores are biased. Although rotation helps us achieve simple structure, if the interrelationships do not hold itself up to simple structure, we can only modify our model. The elements of the Component Matrix are correlations of the item with each component. The rightmost section of this table shows the variance explained by the extracted factors after rotation. The rotated factor model makes some small adjustments to factors 1 and 2, but factor 3 is left virtually unchanged. Factor Scores Method: Regression. We do this using the Harvard and APA styles. The definition of simple structure is that in a factor loading matrix: The following table is an example of simple structure with three factors: Lets go down the checklist of criteria to see why it satisfies simple structure: An easier set of criteria from Pedhazur and Schemlkin (1991) states that. Recall that the goal of factor analysis is to model the interrelationships between items with fewer (latent) variables. Equivalently, since the Communalities table represents the total common variance explained by both factors for each item, summing down the items in the Communalities table also gives you the total (common) variance explained, in this case, $$ (0.437)^2 + (0.052)^2 + (0.319)^2 + (0.460)^2 + (0.344)^2 + (0.309)^2 + (0.851)^2 + (0.236)^2 = 3.01$$. However, before we introduce you to this procedure, you need to understand the different assumptions that your data must meet in order for PCA to give you a valid result. Looking at the Pattern Matrix, Items 1, 3, 4, 5, and 8 load highly on Factor 1, and Items 6 and 7 load highly on Factor 2. Click and Get a FREE Quote. Looking more closely at Item 6 My friends are better at statistics than me and Item 7 Computers are useful only for playing games, we dont see a clear construct that defines the two. In the factor loading plot, you can see what that angle of rotation looks like, starting from \(0^{\circ}\) rotating up in a counterclockwise direction by \(39.4^{\circ}\). Note that there is no right answer in picking the best factor model, only what makes sense for your theory. If these variables are highly correlated, you might want to include only those variables in your measurement scale (e.g., your questionnaire) that you feel most closely represent the construct, removing the others; (b) you want to create a new measurement scale (e.g., a questionnaire), but are unsure whether all the variables you have included measure the construct you are interested in (e.g., depression). For both methods, when you assume total variance is 1, the common variance becomes the communality. If you are looking for help to make sure your data meets assumptions #2, #3, #4 and #5, which are required when using PCA, and can be tested using SPSS Statistics, we help you do this in our enhanced content (see our Features: Overview page to learn more). PCA analyzes the entire correlation matrix . First go to Analyze Dimension Reduction Factor. The figure below shows the Structure Matrix depicted as a path diagram. The sum of the squared eigenvalues is the proportion of variance under Total Variance Explained. The steps to running a two-factor Principal Axis Factoring is the same as before (Analyze Dimension Reduction Factor Extraction), except that under Rotation Method we check Varimax. For example, for Item 1: Note that these results match the value of the Communalities table for Item 1 under the Extraction column. The process of conducting a Principal Components Analysis A Principal Components Analysis) is a three step process: 1. In the genetic data case above, I would include the first 10 principal components and drop the final three variables from Z*. The difference between the figure below and the figure above is that the angle of rotation \(\theta\) is assumed and we are given the angle of correlation \(\phi\) thats fanned out to look like its \(90^{\circ}\) when its actually not. Lets suppose we talked to the principal investigator and she believes that the two component solution makes sense for the study, so we will proceed with the analysis. Factor Scores Method: Regression. If you want to use this criterion for the common variance explained you would need to modify the criterion yourself. Kaiser normalizationis a method to obtain stability of solutions across samples. To get the first element, we can multiply the ordered pair in the Factor Matrix \((0.588,-0.303)\) with the matching ordered pair \((0.773,-0.635)\) in the first column of the Factor Transformation Matrix. Looking at the Rotation Sums of Squared Loadings for Factor 1, it still has the largest total variance, but now that shared variance is split more evenly. Suppose you wanted to know how well a set of items load on eachfactor; simple structure helps us to achieve this. 3. Transfer all the variables you want included in the analysis (. \end{eqnarray} Just as in orthogonal rotation, the square of the loadings represent the contribution of the factor to the variance of the item, but excluding the overlap between correlated factors. Promax really reduces the small loadings. If you look at Component 2, you will see an elbow joint. Information Technology at Procter & Gamble At the end of these 18 steps, we show you how to interpret the results from your PCA. Some criteria say that the total variance explained by all components should be between 70% to 80% variance, which in this case would mean about four to five components. Since this is a non-technical introduction to factor analysis, we wont go into detail about the differences between Principal Axis Factoring (PAF) and Maximum Likelihood (ML).