Try plotting the data. A histogram might help. An x-y plot of one variable against another might also be useful.

Dear Shiela, I would recommend carrying out a principal component analysis on each scale, extracting only one component and saving the regression model for each scale. You can then compare the mean values of each regression model.

How were the original items scored? If they do not give you any measure of an underlying frequency, but are simply ordinal (e.g., "often" versus "seldom"), then you should rely on percentages. For example, which items received the most ratings for "often."

In general, the purpose of collecting these kinds of variables is to create scales that can be used as continuous variables in statistics such as t-tests, correlations, etc. As such, they do not produce data on frequencies.

Your choices for data analysis might be limited if your sample size is small. The principal component analysis might not be very useful if all you want to know is the relative frequency of the three Likert variables. You will have one (maybe two) principal components an you only started with three variables. On the other hand, it might be very useful if you use the 60 Likert-type variables in an effort to improve the model. You are then using the raw data to try and arrive at a small number of key variables, and it is possible that PCA will simply confirm that your 60 Likert-type variables are best described using the three Likert variables that you already have.

You also need to consider your goals. A large questionnaire offers many options for analysis. If one of the questions was "what subject do you teach" then a natural question is to wonder how the three Likert variables differ between teachers of different subjects. However, with only three variables standard MANOVA might be sufficient.

I think the best use of factor analysis would be to examine whether the 60 variables do indeed match up with the three hypothesized factors, but that would not answer the "frequency" question.

Also, I would not recommend principal components for such a factor analysis, because it makes too many assumptions that are seldom met in social science data. Instead, I would suggest using maximum likelihood factor analysis with an oblique rotation (to examine correlated factors).

 I think factor analysis will be better option. Variables can be reduced in some factors and again data can be collected against those factors for further analysis. It needs to be explored from statisicians, where factor loadings of each factors can be fit for further parametric analysis like regression model. 

I'm not sure why people are suggesting factor analysis and PCA. Maybe I am missing something. The questioner is NOT asking about the structure of these 60 items. They are 3 sets for 20 behaviors. 

If you are interested in frequency AND you believe that the item scales match onto frequencies in the same way for different items (e.g., they are numeric labels), and you just want to know which has the highest frequency add the numeric values up. If you are not able to add them up because of the way the scales are constructed (like if labels are things like "a little", "often," then you should probably not be asking about differences among items. But, if you make lots of assumptions, then just saying which has the highest mean of the three groups might be informative.

The fact is that Factor Analysis has the capability of  reducing different variables to a fewer number of variables without loosing any significant part of the original variables. 

However, some of those new variables (factors )  are still correlated with each other.  They are not perfectly independent new variables.  The function of the PCA (Principal Component Analysis)  is to maximally rotate those new variables or factors into orthogonal directions so that they will not longer be correlated. The beauty of this is that the scores from these  new variables are now continuous or interval measurement variables. They can be treated  the  way other continuous variables are treated. You can also recode them into ordinal data or even categorical data and select  the frequencies you want.   A lot of literature already exist on this. (See Harris, Richard  1975; See also the  SPSS manuals.

I personally have developed several practically proven  models using this beautiful PCA.  feature.

@Shiela,

Which answering categories do you have and are they identical for all items? Depending on the categories used there are different answers how to do what you want. If they are binary, you could count the max score for each of the sets and the set with the max score is the preferred one. It may even be the case the the preferred one differs for each teacher. Count the max score can be used with other categories, but might be dangerous if the categories are not well balanced.

If  you have different categories for different sets you might need to show theoretically that the categories are equivalent, but that is not always clear.  In the past I have done this twice with the help of Ivo Molenaar (publications were business secrets,  have never made public and have been shredded). Besides theoretically show they are equivalent we had to 'prove' they were indeed practically identical. 

If you can do all of this, you might want to do:

COUNT SCALE1 = item1 item2 item3 (4) .

COUNT SCALE2 = item4 item5 item6 (4) .

COUNT SCALE3 = item7 item8 item9 (4) .

The Scalex with the highest average is mostly preferred. You can also calculate the preferance for each teacher.

IF (Scale1 > Scale2 and Scale1 > Scale3 ) Pref = 1 .

IF (Scale2 > Scale1 and Scale2 > Scale3 ) Pref = 2 .

IF (Scale3 > Scale1 and Scale3 > Scale2 ) Pref = 3 .

I hope this might help you achieve what you are looking for.

The question asked by Shiela is quite straightforward. What is required is which of three teacher practice  options is the most frequent.  This connotes dominance over others.

Therefore, if Principal Components Analysis is applied on the  entire data set,  the components that will emerge will be displayed in the order of the magnitudes of their eigenvalues which are their variances.  The largest will be displayed first and the last will come last. In fact  PCA  will.also confirm whether there are exactly three components or groups.  The objective is to prioritise the components in the question asked. 

In answer to Daniel Wright's comments about factor analysis, I agree that it isn't relevant to determining frequency. However, it may be unwise to simply assume that the items fall into the 3 predetermined categories, and this is what factor analysis can assess.

The author does seem to want to question the category membership. Suppose the three categories are what are done in math class, in biology class, and in art class.  How will the covariance matrix change whether an action done in biology class should be classified as done in art class? I am not questioning whether for some research questions people might want to pre-suppose some latent variables, but I don't understand why this question is being interpreted as one of these. I think most of us see "60 item scale" and our brains jump into ``dimension reduction'' mode, but this is a much simpler question (albeit depending on what the response alternatives are and some assumptions the answer may be difficult, but we don't know the response alternatives).

The way I read the question is this: I gave a large number of teachers a 60 question survey. I have taken the results from this survey and 20 of the variables are combined to generate 1 Likert variable. I take 20 different answers and generate a second Likert variable. I take the final 20 questions and calculate the third Likert variable. A teacher's time can be devoted to one of these tasks, and I want to know the relative proportion each variable contributes to defining how teachers spend their time.

1) One can wonder how we arrived at these Likert variables, and if there might not be some better way of reducing the original 60 Likert-type items into a small number of Likert variables.

2) We do not know if there were any other questions asked. Department, subject, university, sex, income, years teaching, willingness to take a long survey, and other factors might influence our perception of teaching methods and time allocation based on survey results.

3) We do not know sample size. We might all rethink our answers if the sample size is 10.

Maybe a bar-and-whisker chart is all that is required. It shows the three factors on the x-axis, and their magnitude on the y-axis. While this answers the question as asked, it assumes that the grouping of the 60 questions into 3 sets of 20 was appropriate.

All the question author says is "three main components of teacher practice," so I don't think the rest of us can judge what those did or did not measure.

You're right, I guess we'll have to wait for clarification before trying to answer this one!

Principal Components Analysis has the capability to arrange categories or components in order of descending variance or eigenvalues which represent their weights or importance or dominance. The frequency issue in the question is surrogate to the word dominance.