Don't hesitate to plot the data, which is often much more illuminating than any test.

Regarding PCA, just run it and see if the first principal axis is useful for scoring / ranking (use the PCA scores = coordinates in the new basis) for this purpose). It might be the case if the first eigen value is dominant wrt the 2 others. I also saw situations where it was still interesting even though it was not dominant.

Thank you for your response sir. Actually my data is survey data and it's rating based ( for example 1-5) . I have taken mean of the responses and in my final data set values are between 1-5 , with some decimal values as well. Does it make sense to check non linearity with such data set ? If yes then apart from plotting , is there any statistical test ? 

PCA is looking for an orthogonal basis to project the initial data on, with the first axis rxplaining as much variance as possible, and the second axis orthogonal to the first axis and explaining as much as possible of the residual variance, and so on. In that sense, it is a linear model. However, it is quite robust to non linearities and correlations between the variables. In other words you can use a combination of multivariate plotting techniques (eg the function "pairs" in R), and PCA to explore your dataset.

see https://www.researchgate.net/publication/312491269_Drivers_of_Rift_Valley_fever_epidemics_in_Madagascar, section "Supporting information" for an example of implementation.

Article Drivers of Rift Valley fever epidemics in Madagascar

Hello Anjali. I think you should check multiple correspondance analysis for your data (on original values not means). Mean values are only adequate for real continuous variables.

Thanks for response Jad, however, there are lot of missing values in the data set. Like there are 20 stakeholders which gave their opinion on  20 technologies on the basis of 5 criteria. But say for 10% technologies , all stakeholders have given ratings from 1-5, however for others, very few have given their opinions. In short data is not uniform (like not all stakeholders/participants gave votes for all technologies (whether 1 or 5 ), hence lot of missing data. Thus I took mean response. What do you suggest ? 

One more question:

Now I am trying to run PCA on original data set. The structure is like this : 

                           Criteria A Criteria B Criteria C

Technology A      20 respondents gave rank (1-5) for all 3 criteria 

Technology B      10 respondents ranked only for first 2 , 10 respondents for last 2                                       criteria (hence missing values)

Technology C      Similarly in this -missing values

I am planning to combine all complete information under the 3 criteria, and run pca/correspondence analysis . Sounds fine ?

Please people, provide me with your valuable feedback. 

Going back to the initial question, I would just do a PCA with the respondents as the rows, and criteria as the colums (thus dropping observations with missing values). After ranking / scoring the respondents, I woud add the technology information, and use a rank test or an anova to assess possible differences between technologies.

Regarding missing data, you can use only the 2 first criteria to limit the problem.

Here is an example with dummy data, implemented in R

> ## dummy data

> m1 c1 m2 c2 m3 c3

> dfr head(dfr)

techno c1 c2 c3

1 1 5 3 4

2 1 4 5 1

3 1 3 3 3

4 1 4 2 5

5 1 4 3 4

6 1 3 1 2

>

> ## PCA

> pca barplot(pca$eig)

>

> ## get the scores on the first axis

> dfr$score

> ## anova to assess score difference among techno

> fm summary(fm)

Call:

lm(formula = score ~ techno, data = dfr)

Residuals:

Min 1Q Median 3Q Max

-2.2519 -0.5763 0.1290 0.5033 1.8176

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -0.53875 0.19066 -2.826 0.00756 **

techno2 0.07848 0.33024 0.238 0.81346

techno3 2.07652 0.33024 6.288 2.56e-07 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.8527 on 37 degrees of freedom

Multiple R-squared: 0.5399, Adjusted R-squared: 0.5151

F-statistic: 21.71 on 2 and 37 DF, p-value: 5.782e-07

Hello,

Personally, I don't think using PCA which assumes linearly correlated continuous data , neither using the averages are suitable for ranked  categorical data (this is why Non parametric statistic were invented). I would use Correspondance analysis for your data (Burt Table decompostion instead of covariances or correlations decomposition). The type of Technology can be added as an extra categorical variable. MIissing values rows are omited from the analysis.

Well, you are right regarding non-linearities and PCA, though I still believe, from experience, that the method can be used on tables of scores. In practice, with a low-dimension table like here, I doubt non-linearity is a big issue. To be checked. Moreover, it depends on the kind (shape) of the non-linearity. In some cases, ranking/scoring is not affected by  non-linearity.

Also, correspondence analysis is for contingency tables (tables of counts) which is not the case here. It should be possible to use multiple vorrespondence analysis, with the drawback of losing the ordering of categories (=marks) wrt the original score. At last, the overall variance is "diluted" into many axes, and the barplot of eigen values rarely shows a dominant eigen value, making scoring/ranking less obvious.

So, as always, one has to find the best trade-off between imperfect solutions ;-)

Iam quite confused now :-( 

Just explaining briefly again: 

I have 43 technologies ranked between -5 to 5 on the basis of 3 criteria. A total of 20 respondents participated in this survey,however not all gave their response for all technologies. Like all 20 ranked technology A from -5 to 5 against all 3 criteria. So I have data for all 20 respondents. But for technology B, only 10 gave their ranking,but say for only 2 criteria, not for third. Hence, this kind of missing value pattern is in my dataset. I have removed all missing values, so after that,in my data set,I have 20 respondents ranking for technology A but for technology B I now have only 5 respondents. 

Using this complete information dataset, I want to arrive at a composite score for each technology. With pca, I just checked KMO and Bartlett Test, and arrived at an index using first principal component. After that I switched to my mean response data ( 43 points for 43 technologies). Using the index (linear combination) derived from my PCA, I calculated a composite score for every technology. Renaud Sir, is it okay to do so? 

My second question is to Jad Sir. Can I get such a composite score from correspondence analysis for every technology ? Also, correspondence analysis works on frequency data, so since for technology A I have 20 respondents views, but for technology B only 5 , will it not create a biased result? 

Please guide. And thanks again for your diligent participation. 

@Anjali: I think so. If you are based in Montpellier, we can organise a face-to-face meeting for better interaction.

Regarding CoA (or multiple correspondence analysis), the bias also occurs for PCA or any other technique, but what else can you do ?

Concerning your question about correspondence analysis, yes in theory you can have scores per type of technology by applying the analysis independently for each technology (similarly to SIMCA-PCA method if you are familiar with it). My concern is that I’m not sure you have enough respondent to do this...

If you plan to apply correspondence analysis, make sure to organize first your data in binary format; that is, each variable is transformed to n binary variables with n=number of categories of each variable.

If you leave your data in its original form, the multiple correspondence analysis will only calculate the coordinates for the columns (loadings), in binary form you will also get the rows coordinates (scores).

@Renaud Sir , I am based in New Delhi (India) ,so I guess that will not be possible. Thanks again for your cooperation. 

I am attaching scatter plots of my three criteria with this file. To me, there seems no problem of any non-linearity. Your input will be really appreciated. I guess if there is no non-linearity , I can proceed with PCA for this data set. 

@ Jad Sir, could you suggest some useful link for a similar type of query? 

And you suggested applying Correspondence analysis for every technology separately? Actually for some technologies, only 1 respondent has given his preferences. So honestly, I am not able to understand how to fit correspondence analysis to attain my goal. Perhaps, for me it is a conceptual problem first. Anyways, I am digging further, to see how I can make a use of it. 

Thank you so much sir again for your cooperation. 

 Also, @Jad Sir, I simply tried MCA in Stata, using only data variable of three criteria and not its corresponding technologies. So in the loading plot,I could see only the numbers -5 to 5 with different colours, These colours depicted my three criteria. However, I don't think such results will be useful for my study. 

For non-linearity check, I used simple regression between criteria A and C. I regressed A on C, and the t value wasn't significant. I again regressed it on its square, surprisingly it came significant. From scatter-plot I couldn't identify, however from such regression results I suspect some non-linearity , which seems difficult to explain (since it is a rating dataset). Hence, I guess PCA will not be possible here.

Can this help: https://www.researchgate.net/publication/319469038_New_Interpretation_of_Principal_Components_Analysis?

Article New Interpretation of Principal Components Analysis