Feb 27, · Exploratory Essay Uses. Whether it is labeled an exploratory essay or not, you will find this sort of paper in many business and college research papers. The basic point of this paper is to let you examine all the different viewpoints on an issue. Here are some examples of exploratory questions For example, \(\) is the effect of Factor 1 on Item 1 controlling for Factor 2 and \(\) is the effect of Factor 2 on Item 1 controlling for Factor 1. 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 Wonderful Ways. Exploratory research is a very versatile research blogger.com can help you find faults in your case study or even your marketing blogger.com research can lead to further investigations like qualitative and descriptive research
Principal Components (PCA) and Exploratory Factor Analysis (EFA) with SPSS
This seminar will give a practical overview of both principal components analysis PCA and exploratory factor analysis EFA using SPSS. We will begin with variance partitioning and explain how it determines the use of a PCA or EFA model.
For the PCA portion of the seminar, we will introduce topics such as eigenvalues and eigenvectors, communalities, sum of squared loadings, total variance explained, and choosing the number of components to extract. For the EFA portion, we will discuss factor extraction, estimation methods, factor rotation, and generating factor scores for subsequent analyses. The seminar will focus on how to run a PCA and EFA in SPSS and thoroughly interpret output, using the hypothetical SPSS Anxiety Questionnaire as a motivating example.
The basic assumption of factor analysis is that for a collection of observed variables there are a set of underlying or latent variables called factors smaller than the number of observed variablesthat can explain the interrelationships among those variables. Click on the preceding hyperlinks to exploratory essay example the SPSS version of both files. The SAQ-8 consists of the following questions:.
Due to relatively high correlations among items, this would be a good candidate for factor analysis. Recall that the goal of factor analysis is to model the interrelationships between items with fewer latent variables. These interrelationships can be broken up into multiple components. Since the goal of factor analysis is to model the exploratory essay example among items, we focus primarily on the variance and covariance rather than the mean.
Factor analysis assumes that variance can be partitioned into two types of variance, exploratory essay example, common and unique. The total variance is made up to common variance and unique variance, and unique variance is composed of specific and error variance. Here you see that SPSS Anxiety makes up the common variance for all eight items, but within each item exploratory essay example is specific variance and error variance. Now that we understand partitioning of variance we can move on to performing our first factor analysis.
In fact, exploratory essay example, the assumptions we make about variance partitioning affects which analysis we run. As a data analyst, the goal of a factor analysis is to reduce the number of exploratory essay example to explain and to interpret the results.
This can be accomplished in two steps:. Factor extraction involves making a choice about the type of model as exploratory essay example the number of factors to extract.
Factor rotation comes after the factors are extracted, with the goal of achieving simple structure in order to improve interpretability. There are two approaches to factor extraction which stems from different approaches to variance partitioning: a principal components analysis and b common factor analysis.
Unlike factor analysis, exploratory essay example, principal components analysis or PCA makes the assumption that there is no unique variance, the total variance is equal to common variance.
Recall that variance can be partitioned into exploratory essay example and unique variance, exploratory essay example.
If there is no unique variance then common variance takes up total variance see figure below. Additionally, if the total variance is 1, exploratory essay example, then the common variance is equal to the communality.
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. Although the following analysis defeats the purpose of doing a PCA we will begin by extracting as many components as possible as a teaching exploratory essay example and so that we can decide on the optimal exploratory essay example of components to extract later, exploratory essay example.
First go to Analyze — Dimension Reduction — Factor. Move all the observed variables over the Variables: box to be analyze. Under Extraction — Method, pick Principal components and make sure to Analyze the Correlation matrix. We also request the Unrotated factor solution and the Scree plot. Under Extract, choose Fixed number of factors, and under Factor to extract enter 8.
We also bumped up the Maximum Iterations of Convergence to Eigenvalues represent the total amount of variance that can be explained by a given principal component. They can be positive or negative in theory, but in practice exploratory essay example explain variance which is always positive. 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.
Eigenvectors represent a weight for each eigenvalue. The eigenvector times the square root of the eigenvalue gives the component loadings which can be interpreted as the correlation of each item with the principal component. We can calculate the first component as. The components can be interpreted as the correlation of each item with the component. Each item has a loading corresponding to each of the 8 components, exploratory essay example.
This is also known as the communalityand in a PCA the communality for each item is equal to the total variance. 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.
For example, to obtain the first eigenvalue we calculate:. Recall that the eigenvalue represents the total amount of variance that can be explained by a given principal component. Starting from the first component, each subsequent component is obtained from partialling out the previous component.
Therefore the first component explains the most variance, and the last exploratory essay example explains the least. Looking at the Total Variance Explained table, you will get the total variance explained by each component.
Because we extracted the same number of components as the number of items, the Initial Eigenvalues column is the same as the Extraction Sums of Squared Loadings column. Since the goal of running a PCA is to reduce our set of variables down, it would useful to have a criterion for selecting the optimal number of components that are of course smaller than the total number of items. One criterion is the choose exploratory essay example that have eigenvalues greater than 1.
Under the Total Variance Explained table, we see the first two components have an eigenvalue greater than 1. This can be confirmed by the Scree Plot which plots the eigenvalue total variance explained by the component number. Recall that we checked the Scree Plot option under Extraction — Display, so the scree plot should be produced automatically. The first component will always have the highest total variance and the last component will always have the least, but where do we see the largest drop?
Using the scree plot we pick two components. Picking exploratory essay example number of components is a bit of an art and requires input from the whole research team. Running the two component PCA is just as easy as running the 8 component solution.
The only difference is under Fixed number of factors — Factors to extract you enter 2. We will focus the differences in the output between the eight and two-component solution. Under Total Variance Explained, we see that the Initial Eigenvalues no longer equals the Extraction Sums of Squared Loadings. Again, we interpret Item 1 as having a correlation of 0.
From glancing at the solution, exploratory essay example, we see that Item 4 has the highest correlation with Component 1 and Item 2 the lowest. Similarly, we see that Item 2 has the highest correlation with Component 2 and Item 7 the lowest.
The communality is the sum of the squared component loadings up to the number of components you extract. In the SPSS output you will see a table of communalities. Since PCA is an iterative estimation process, it starts with 1 as an initial estimate of the communality since this is exploratory essay example total variance across all 8 componentsand then proceeds with the analysis until a final communality extracted. Notice that the Extraction column is smaller than the Initial column because we only extracted two components.
Recall that squaring the loadings and summing down the components columns gives us the communality:. Is that surprising? In an 8-component PCA, how many components must you extract so that the communality for the Initial column is equal to the Extraction column? F, the eigenvalue is the total communality across all items for a single component, 2. F you can only sum communalities across items, and sum eigenvalues across components, exploratory essay example, but if you do that they are equal, exploratory essay example.
The partitioning of variance differentiates a principal components analysis from what we call common factor analysis. Both methods try to reduce the dimensionality of the dataset down to fewer unobserved variables, but whereas PCA assumes that exploratory essay example common variances takes up all of total variance, common factor analysis assumes that total variance can be partitioned into common and unique variance. It is usually more reasonable to assume that you have not measured your set of items perfectly.
The unobserved or latent variable that makes up common variance is called a factorhence the name factor analysis. The other main difference between PCA and factor analysis lies in the goal of your analysis. If your goal is to simply reduce your variable list down into a linear combination of smaller components then PCA is the way to go. However, if you believe there is some latent construct that defines the interrelationship among items, then factor analysis may be more appropriate. In this case, we assume that there is a construct called SPSS Anxiety that explains why you see a correlation among all the items on the SAQ-8, we acknowledge however that SPSS Anxiety cannot explain all the shared variance among items in the SAQ, so we model the unique exploratory essay example as well.
Based on the results of the PCA, we will start with a two factor extraction. To run a factor analysis, use the same steps as running a PCA Analyze — Dimension Reduction — Factor except under Method choose Principal axis factoring, exploratory essay example. Note that we continue to set Maximum Iterations for Convergence at and we will see why later.
We will get three tables of output, Communalities, Total Variance Explained and Factor Matrix. The most striking difference between this communalities table and the one from the PCA is that the initial extraction is no longer one. 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.
To see this in action for Item 1 run a linear regression where Item 1 is the dependent variable and Items 2 -8 are independent variables. Go to Analyze — Regression — Linear and enter q01 under Dependent and q02 to q08 under Independent s. Note that 0. We can do eight more linear regressions in order to get all eight communality estimates but SPSS already does that for us. Like PCA, factor analysis also uses an iterative estimation process to obtain the final estimates under the Extraction column.
Finally, exploratory essay example, summing all the rows of the extraction column, and we get 3. This represents the total common variance shared among all items for a two factor solution. The next table we will look at is Total Variance Explained. In fact, exploratory essay example, SPSS simply borrows the information from the PCA analysis for use in the factor analysis and the factors are actually components in the Initial Eigenvalues column.
The main difference now is in the Extraction Sums of Squares Loadings. We notice that each corresponding row in the Extraction column is lower than the Initial column.
Expository Essay Example - Step by step
, time: 9:29Exploratory Papers // Purdue Writing Lab
Exploratory essays ask questions and gather information that may answer these questions. However, the main point of the exploratory or inquiry essay is not to find definite answers. The main point is to conduct inquiry into a topic, gather information, and share that information with readers. Introductions for Exploratory Essays The real reason there is so much crime is down to problems such as poor living conditions, unemployment, and so on (here, “such as” can be interchanged for “for instance/for example.”) Writing this type of essay is not as difficult as it might at first seem Disclaimer: The Reference papers provided by the Students Assignment Help serve as model and sample papers for students and are not to be submitted as it is. These papers are intended to be used for reference and research purposes only. Students Assignment Help rated /5 based on + customer reviews
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