QUESTION 4. (20 marks) Consider the raw dataset below with 12 observations, on 5 socio-economic variables, called Population, School, Employment, Services and HouseValue. data SocioEconomics; input Population School Employment Services HouseValue; datalines ; 5700 12. 8 2500 270 25000 1000 10.9 600 10 10000 3400 8. 8 1000 10 9000 3800 13.6 1700 140 25000 4000 12. 8 1600 140 25000 8200 8 . 3 2600 60 12000 1200 11 . 4 400 10 16000 9100 11.5 3300 60 14000 9900 12.5 3400 180 18000 9600 13.7 3600 390 25000 9600 9.6 3300 80 12000 9400 11.4 4000 100 13000 proc factor data SocioEconomics simple corr; run; Conduct a factor analysis by using the SAS statements above. Show your SAS code (it can vary to the one I suggest), output and answers for the following questions: 1. Prepare the dataset for a Factor analysis via SAS. (1 mark) 2. Generate the means and standard deviations of the data. (1 mark) 3. Perform a Factor analysis on the raw data AND the correlation matrix using the code above and answer the following questions. (6 marks) 4. From the eigenvalues of the correlation matrix and the factor loading matrix and communalities outputted answer the following questions. a) Do the first two principal components (factors) provide an adequate summary of the data? (1 mark) b) How much of the variation is accounted for by 2 factors? (1 mark) c) How much of the variation is accounted for by 3 factors? (1 mark) 5. To get the scoring coefficients as eigenvalues use PROC PRINCOMP below, proc princomp data SocioEconomics ; run ; Now use this output from SAS princomp to answer the following questions: a) what are the eigenvalues and the respective eigenvectors? (3 marks) ) What is the proportion of the variance accounted for by the first and second component respectively? (1 mark) ) Together how much do the first and second factors together account for the standardised variance? (1 mark) d) Do the final communality estimates show that all the variables are well accounted for by how many components or factors? Justify your answer. (1 mark) 6. To obtain the component scores as linear combinations of the observed variables request the standardized scoring coefficients by adding the SCORE option in the FACTOR statement: and run this. Note that the SCORE option in the code below requests the display of the standardized scoring coefficients. proc factor data=SocioEconomics n=5 score; run ; As each factor/component can expressed as a linear combination of the standardised observed variables using the code above, answer the following questions: a) Write down the first principal component or Factor in terms of the standardised variables. (1 mark) b) Write down the second principal component or Factor2 in terms of the standardised variables. (1 mark) c) Write the first and second PCs in terms of eigenvectors. (1 mark) NOTES/HINTS: The SIMPLE option specified in the PROC FACTOR statement generates the means and standard deviations of all observed variables in the analysis The CORR option specified in the PROC FACTOR statement generates the output of the observed correlations. To express the observed variables as functions of the components (or factors), you inspect the factor loading matrix. To obtain the component scores as linear combinations of the observed variables request the standardized scoring coefficients by adding the SCORE option in the FACTOR statement: The SCORE option in the code below requests the display of the standardized scoring coefficients proc factor data SocioEconomics n=5 score; run