Question

Run a Principal Axis Factor Analysis (PFA), with default settings for the “Eigenvalue greater than 1” option. Just use the default rotation of ‘orthogonal’ for now.

a.)Considering at least two rules-of-thumb discussed, what do the eigenvalues (you will need to “eyeball” these values from the plots) and scree plot suggest about the number of factors underlying the variables? Explain your answer (10 points)

b.)Interpret and explain the communality variables (Hint: remember communality = 1 – uniqueness)? In what way do they reflect something that the items’ have in common? Are there any variables which have low communalities (said different, high uniqueness) and, if so, what does this suggest (5 points)?

https://montclair.instructure.com/courses/92554/files/5353499/download?verifier=PWcv383xNlOVhmZPkaaWjzgFNS4ibD6JO6X6upRe&wrap=1

https://montclair.instructure.com/courses/92554/files/5353534/download?verifier=8tmhZOC0R40nJD36XXG86cUZI8u3FvaIC4g2yGTg&wrap=1

Answer 1

For given data set with 9 variables, run PFA with orthogonal rotation and 'eigen value greater than 1' rule.

a) Eigenvalues used to decide number of factors underlying the variables. Principal components with high eigenvalues are chosen.

First we will use rule of thumb to getting principal components that eigenvalue greater than 1.

In above output, there are two components those who have eigen values greater than 1. Hence we we will select two factors with help of "eigenvalue greater than 1" rule. Here 50% variation is contributed by first two components(which is slightly low).

Scree plot is graph of eigen value vs components.

In this scree plot we can see that eigen values show straight line with third component. when line gets flat then we can say that remaining components constribute less to the variation. Hence after second component there is flat line.

Hence both method suggest that we can use 2 factors.

b) Communality is the variance which is explained by the each components. It is also defined as sum of squares of factor loadings. Initial value of communality is 1. Please refer to the following table for communality matrix.

Variables with high comminality value can be presented in common factor space. If all variables have high communality values then we can say that all variables are represented well.

However variables with low communality values are not represented well. If communality is low then we need to extract another components.

In this case, there are some variables which have low communalities that is some variables have extraction value less than 0.5. When there are low communality value then variables are not represented well.