The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Why eigenvalues are greater than 1 in factor analysis? Ask Question. Asked 8 years, 1 month ago. Active 7 years, 10 months ago. Viewed 29k times. Improve this question. Add a comment. Active Oldest Votes. Improve this answer. Peter Flom Peter Flom Factors can only be a linear combination of the observed variables if you "flip the arrows" i.
Girden, I paraphrase like so: because you don't want to end up with more or equal number of factors as the number of your variables is. Ehsan88 Ehsan88 1 1 gold badge 4 4 silver badges 11 11 bronze badges. Likewise, do the variables that are loading on different factors measure something different? Keep in mind that each of the identified factors should have at least three variables with high factor loadings, and that each variable should load highly on only one factor.
After looking at the scree plot as a guide, I often wind up forcing my analysis to run between one and five factors, and then develop the five models separately. Usually it quickly becomes clear when to drop a factor solution, especially when one factor has only two important variables and therefore does not explain much of the overall variability, or if it is not very convincing based on my theoretical expectations.
About the Author: Maike Rahn is a health scientist with a strong background in data analysis. Maike has a Ph. Hey, I want to ask you about the scree plot a little more… If a scree graph is given then how should you interpret it? Thank you! Anything less suggest high degree of multicollinearity which implies that there are variables with high coefficient correlation with other variables. You need to delete some of these variables from the model and ensure the determinant is higher than 0.
Look at the correlation matrix to spot high correlation coefficient values of more than 0. In what order would the questions be?
Random Selection C. Hi, Thanks for sharing the knowledge. If I want to cite, what should I write? God bless. Thanks, I really found this write-up very helpful. I am about to conduct Factor Analysis to establish the construct validity of my data collection instruments.
So, this article has widen my horizons on factor analysis. Thanks a lot. Hi, I am a little bit confused about factor analysis. I would like to ask some questions. Because the results of fixed factors are some time good than the above. If we use this fixed factor option in spss, how we can can explain and give reference for it?
May seem like a stupid question, sorry. I understand the math when it comes to determining eigenvalues from a correlation matrix. I understand they are a type of correlation, but how are these numbers generated? Thanks u so much for your contribution of knowledge towards factor analysis, it has been a very good explanation, clear and concise. Though am very new to the topic, I still need more exposition in regards the topic how and when to use it, majorly the interpretation of the screen plot and the uses.
Hi Maike,I am appreciating your contribution for this. I am too new to this field and carrying out one research on job satisfaction where i have used various factors affecting job satisfaction. Those questions were basically on various factors like pay, perks and benefits, administrative policies etc. So can i consider those questions as different factors for factors analysis? Is factor analysis is useful in reducing those 21 questions factors in small number of question factors? In words, this is the total common variance explained by the two factor solution for all eight items.
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. True or False the following assumes a two-factor Principal Axis Factor solution with 8 items. F, the sum of the squared elements across both factors, 3.
F, the total Sums of Squared Loadings represents only the total common variance excluding unique variance, 7. F, eigenvalues are only applicable for PCA. To run a factor analysis using maximum likelihood estimation under Analyze — Dimension Reduction — Factor — Extraction — Method choose Maximum Likelihood. Although the initial communalities are the same between PAF and ML, the final extraction loadings will be different, which means you will have different Communalities, Total Variance Explained, and Factor Matrix tables although Initial columns will overlap.
The other main difference is that you will obtain a Goodness-of-fit Test table, which gives you a absolute test of model fit. Non-significant values suggest a good fitting model. Here the p -value is less than 0. In practice, you would obtain chi-square values for multiple factor analysis runs, which we tabulate below from 1 to 8 factors.
The table shows the number of factors extracted or attempted to extract as well as the chi-square, degrees of freedom, p-value and iterations needed to converge.
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. Practically, you want to make sure the number of iterations you specify exceeds the iterations needed. In SPSS, no solution is obtained when you run 5 to 7 factors because the degrees of freedom is negative which cannot happen.
The number of factors will be reduced by one. It looks like here that the p -value becomes non-significant at a 3 factor solution. Note that differs from the eigenvalues greater than 1 criterion which chose 2 factors and using Percent of Variance explained you would choose factors. We talk to the Principal Investigator and at this point, we still prefer the two-factor solution. We will talk about interpreting the factor loadings when we talk about factor rotation to further guide us in choosing the correct number of factors.
F, the two use the same starting communalities but a different estimation process to obtain extraction loadings, 3. F, only Maximum Likelihood gives you chi-square values, 4. F, greater than 0. T, we are taking away degrees of freedom but extracting more factors. 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.
For both methods, when you assume total variance is 1, the common variance becomes the communality. The communality is unique to each item, so if you have 8 items, you will obtain 8 communalities; and it represents the common variance explained by the factors or components.
However in the case of principal components, the communality is the total variance of each item, and summing all 8 communalities gives you the total variance across all items. 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.
In summary, for PCA, total common variance is equal to total variance explained , which in turn is equal to the total variance, but in common factor analysis, total common variance is equal to total variance explained but does not equal total variance.
The following applies to the SAQ-8 when theoretically extracting 8 components or factors for 8 items:. F, the total variance for each item, 3. F, communality is unique to each item shared across components or factors , 5. After deciding on the number of factors to extract and with analysis model to use, the next step is to interpret the factor loadings. Factor rotations help us interpret factor loadings. There are two general types of rotations, orthogonal and oblique.
The goal of factor rotation is to improve the interpretability of the factor solution by reaching simple structure. Without rotation, the first factor is the most general factor onto which most items load and explains the largest amount of variance. This may not be desired in all cases. Suppose you wanted to know how well a set of items load on each factor; simple structure helps us to achieve this. For the following factor matrix, explain why it does not conform to simple structure using both the conventional and Pedhazur test.
Using the Pedhazur method, Items 1, 2, 5, 6, and 7 have high loadings on two factors fails first criterion and Factor 3 has high loadings on a majority or 5 out of 8 items fails second criterion. We know that the goal of factor rotation is to rotate the factor matrix so that it can approach simple structure in order to improve interpretability. Orthogonal rotation assumes that the factors are not correlated. 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.
The most common type of orthogonal rotation is Varimax rotation. We will walk through how to do this in SPSS. First, we know that the unrotated factor matrix Factor Matrix table should be the same.
Additionally, since the common variance explained by both factors should be the same, the Communalities table should be the same. The main difference is that we ran a rotation, so we should get the rotated solution Rotated Factor Matrix as well as the transformation used to obtain the rotation Factor Transformation Matrix. Finally, although the total variance explained by all factors stays the same, the total variance explained by each factor will be different.
The Rotated Factor Matrix table tells us what the factor loadings look like after rotation in this case Varimax. Kaiser normalization is a method to obtain stability of solutions across samples. After rotation, the loadings are rescaled back to the proper size. This means that equal weight is given to all items when performing the rotation. The only drawback is if the communality is low for a particular item, Kaiser normalization will weight these items equally with items with high communality.
As such, Kaiser normalization is preferred when communalities are high across all items. You can turn off Kaiser normalization by specifying. Here is what the Varimax rotated loadings look like without Kaiser normalization. Compared to the rotated factor matrix with Kaiser normalization the patterns look similar if you flip Factors 1 and 2; this may be an artifact of the rescaling. The biggest difference between the two solutions is for items with low communalities such as Item 2 0.
Kaiser normalization weights these items equally with the other high communality items. In the both the Kaiser normalized and non-Kaiser normalized rotated factor matrices, the loadings that have a magnitude greater than 0.
We can see that Items 6 and 7 load highly onto Factor 1 and Items 1, 3, 4, 5, and 8 load highly onto Factor 2. Item 2 does not seem to load highly on any factor. The figure below shows the path diagram of the Varimax rotation. Comparing this solution to the unrotated solution, we notice that there are high loadings in both Factor 1 and 2. This is because Varimax maximizes the sum of the variances of the squared loadings, which in effect maximizes high loadings and minimizes low loadings.
In SPSS, you will see a matrix with two rows and two columns because we have two factors. How do we interpret this matrix? How do we obtain this new transformed pair of values? The steps are essentially to start with one column of the Factor Transformation matrix, view it as another ordered pair and multiply matching ordered pairs.
We have obtained the new transformed pair with some rounding error. The figure below summarizes the steps we used to perform the transformation. The Factor Transformation Matrix can also tell us angle of rotation if we take the inverse cosine of the diagonal element.
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. This makes sense because if our rotated Factor Matrix is different, the square of the loadings should be different, and hence the Sum of Squared loadings will be different for each factor. However, if you sum the Sums of Squared Loadings across all factors for the Rotation solution,.
This is because rotation does not change the total common variance. 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.
Varimax rotation is the most popular orthogonal rotation. 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. Higher loadings are made higher while lower loadings are made lower. This makes Varimax rotation good for achieving simple structure but not as good for detecting an overall factor because it splits up variance of major factors among lesser ones.
Quartimax may be a better choice for detecting an overall factor. It maximizes the squared loadings so that each item loads most strongly onto a single factor. Here is the output of the Total Variance Explained table juxtaposed side-by-side for Varimax versus Quartimax rotation. You will see that whereas Varimax distributes the variances evenly across both factors, Quartimax tries to consolidate more variance into the first factor. Equamax is a hybrid of Varimax and Quartimax, but because of this may behave erratically and according to Pett et al.
Like orthogonal rotation, the goal is rotation of the reference axes about the origin to achieve a simpler and more meaningful factor solution compared to the unrotated solution. In oblique rotation, you will see three unique tables in the SPSS output:. Suppose the Principal Investigator hypothesizes that the two factors are correlated, and wishes to test this assumption.
The other parameter we have to put in is delta , which defaults to zero. Larger positive values for delta increases the correlation among factors. In fact, SPSS caps the delta value at 0. Negative delta may lead to orthogonal factor solutions. F, larger delta values, 3. The factor pattern matrix represent partial standardized regression coefficients of each item with a particular factor.
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. The figure below shows the Pattern Matrix depicted as a path diagram. Remember to interpret each loading as the partial correlation of the item on the factor, controlling for the other factor.
The more correlated the factors, the more difference between pattern and structure matrix and the more difficult to interpret the factor loadings. Looking at the Factor Pattern Matrix and using the absolute loading greater than 0. In the Factor Structure Matrix, we can look at the variance explained by each factor not controlling for the other factors.
In general, the loadings across the factors in the Structure Matrix will be higher than the Pattern Matrix because we are not partialling out the variance of the other factors. The figure below shows the Structure Matrix depicted as a path diagram.
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