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Which methods could be used in order to reduce dimensionality from n to 1?

Giving an example:

I have a data set of 8 variables, and I would like to obtain one combination that describe the maximum information in order to get a score of the importance of each variable.

Going deeper:

Imagine these 8 variables are related with interaction's web user, so I would like to score each website depending on this iteration(8 variables). So, a way to get this?

As far as I worked with, I know

  • PCA
  • T-SNE
  • FA(Factor Analysis)
  • CFA(Confirmatory Factor analysis)

can be a way to do that.

Which method could be the best one?

EDIT:(EXAMPLE)

Simulating some data..

names=   ('page1','page2','page3')
n_clicks = (30, 1,10)
Time_visit = (100,2,50)
n_likes = (2223,10,100)

With some algotihm we would get the following Scores:

page1_Score = 120; page2_Score = 10; page3_Score = 60

PeCa
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2 Answers2

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I do not know about T-SNE at all, but each of the other three could be the "best one," depending on the assumptions you make on the way your data are generated.

  1. PCA is often used when we believe that the items make up the composite variable. This would be an example like socioeconomic status: Items like education, salary, job prestige, etc., are constituent parts of socioeconomic status.

  2. EFA is often used when we believe that the items are generated from a latent construct. This would be an example like happiness: We cannot physically measure happiness, so we ask people a number of questions that we think are outcomes of how happy people are. Those items don't make up happiness; instead, we assume that happiness is affecting these.

  3. CFA makes the same assumption that EFA does (this is called the "common factor model"). While EFA is unsupervised—it does not assume a structure for the data (i.e., you could get one factor, two factors, three factors, whatever)—a CFA assumes that there is a structure to your data. The researcher specifically says what items load on what factor. You can get model fit statistics that tell you how much this model-implied covariance matrix actually maps onto the real covariance matrix, but you must first start with a theoretical structure in mind.

You say that "I would like to obtain one combination that describe the maximum information," which sounds like PCA is what you want. You also say that you wish to do this "in order to get a score of the importance of each variable." This is tricky. How do you define importance? Is it predictive validity? Then dimension reduction may not be what you are at. But a PCA can tell you how much that item loads—or contributes to—that component.

It depends on your theoretical assumptions for how you think the data were generated and if you have a factor structure already in mind. But it seems to me from the limited information given that you are after a PCA.

I describe more on the difference between PCA and PAF (a type of EFA) here.

Mark White
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    Good explanation, but I'll add a bit of commentary. PCA may identify one or more principal components, like EFA might identify one or more latent factors. PeCa: it sounds like you are assuming that your 8 items are all related to one underlying construct. Is this true? If this is true, you can still run PCA and see how many principal components are identified, but if you identify more than one component, you may want to think about whether or not a single sum score is really justified. – Weiwen Ng May 25 '17 at 15:08
  • Yes, of course. Sorry if I was unclear about PCA only returning one component. Like EFA, there can be multiple. – Mark White May 25 '17 at 15:11
  • The point is to find how to get the proper relation to create a score to rate a website by their visits, clicks etc... I mean, with Data Science tools. One example could be the following: if one website has 30 clicks, 100 minutes of user presence, etc... Some method could give a Score of 120... based on an algorithm.. – PeCa May 25 '17 at 15:18
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To add one more option to your list: you may also consider item response theory (IRT) or multi-dimensional IRT. See Phil Chalmers' site for a good intro, including training materials.

user162760
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