RESEARCH AND SURVEY STATISTICS – STA3022F SOLUTION TO TUTORIAL #2 Week 3 2007 FACTOR ANALYSIS QUESTION 1 Q1.1 METHOD 1: Sum communalities across all variables on Unrotated Factor 1… Eigenvalue1 = 0.8201 + 0.7733 + 0.3766 + 0.4023 + 0.4154 + 0.6785 = 3.4665 METHOD 2: Square and sum factor loadings across all variables on Unrotated Factor 1… Eigenvalue1 = (-0.9056)2 + (-0.8794) 2 + (0.6136) 2 + (0.6343) 2 + (0.6445) 2 + (-0.8237) 2 = 3.4665 Q1.2 Total amount of variation is equal to number of variables in the model = 6. So the amount of variation explained by Unrotated Factor 1 is 3.4665/6 = 0.5777 or 57.77%. Q1.3 You are given the eigenvalue for Factor 2 (= 1.8833), so amount of variation explained by the two significant unrotated factors is (3.4665 + 1.8833)/6 = 0.8916 or 89.16%. Q1.4 Here you need to work out the communalities “From 2 Factors”... Communality from 2 factors = (Loading on Factor 1)2 + (Loading on Factor 2)2 So, for ADDTIME: Comm from 2 factors = 0.90562 + (-0.3953)2 = 0.9765 MULTIME: Comm from 2 factors = 0.87942 + (-0.4479)2 = 0.9739 MINWORD: Comm from 2 factors = (-0.6136)2 + (-0.7206)2 = 0.8959 Similarly for others Communalities (Factor10 - Microprocessor Perf ormance Study ) Extraction: Principal components Rotation: Unrotated From 1 From 2 Multiple Variable Factor Factors R-Square ADDTIME 0.820176 0.976501 0.952772 MULTIME 0.773374 0.973993 0.950067 MINWORD 0.376612 0.895952 0.863507 MAXWORD 0.402348 0.920072 0.873757 CAPACITY 0.415498 0.620947 0.548009 CY CLE 0.678535 0.962384 0.931122
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Note: You could also work it out using Communality from 2 factors = Communality from 1 factor + (Loading on Factor 2)2 Q1.5 Use method 2 from Q1.1: Eigenvalue1 = Square and sum factor loadings across all variables on Unrotated Factor 1 Eigenvalue2 = Square and sum factor loadings across all variables on Unrotated Factor 2 Factor Loadings (Varimax normalized) (Factor10 - Microprocessor Perf ormance Study ) Extraction: Principal components (Marked loadings are >.700000) Factor Factor Variable 1 2 ADDTIME -0.963711 -0.218546 MULTIME -0.973739 -0.160704 MINWORD 0.066928 0.944178 MAXWORD 0.084199 0.955501 CAPACITY 0.250345 0.747177 CY CLE -0.979215 -0.059344 Expl.Var 2.910011 2.439839 Prp.Totl 0.485002 0.406640
For Unrotated factors, proportion explained by factor 1 was 57.77% and proportion explained by factor 2 was 31.38%. Some of the variation that was being explained by factor 1 has been reallocated and is now being explained by factor 2. Factor 1 has thus given up some of its explanatory power to Factor 2, with the expectation that this leads to better interpretations. Q1.6 F a c to r L o a di n g s , F a c to r 1 v s . F ac to r 2 R o ta ti o n : V a rim a x n o rm a l i z e d E x t r a c t io n : P r i n c i p a l c o m p o n e n ts 1 .2 1 .0
MM IAN XW WO OR RD D
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Q1.7 Factor 1: Speed of microprocessor (ADDTIME, MULTIME, CYCLE) Factor 2: Storage capability (MINWORD, MAXWORD, CAPACITY)
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QUESTION 2 Q2.1 From rotated factor loadings: Communality of variable B3 (PDT TESTIN) = 0.60262 + 0.08792 + 0.70532 = 0.8683 i.e. 86.83% of the variation in B3 (Product testing) is explained by the 3 significant factors. Q2.2 This question is ambiguous because it doesn’t specify whether to use the ROTATED or UNROTATED factors (this should have been specified in the question) For UNROTATED factors: Total amount of variation = Number of independent variables = 7 Factor 1: % Explained variation = 2.9514/7 = 42.16% Factor 2: % Explained variation = 1.4441/7 = 20.63% Factor 3: % Explained variation = 1.0202/7 = 14.57% For ROTATED factors: First need to get eigenvalue for each factor Factor 1 is given as 2.1518 For factors 2 and 3, use method 2 from Q1.1: Eigenvalue2 = Square and sum factor loadings across all variables on Rotated Factor 2 = 0.00922 + 0.95352 + 0.08792 + 0.32932 + 0.05702 + 0.87972 + 0.09752 = 1.8123 Eigenvalue3 = Square and sum factor loadings across all variables on Rotated Factor 3 = 0.96502 + 0.04022 + 0.70532 + 0.05892 + 0.00762 + 0.01882 + 0.13232 = 1.4517 Then, total amount of variation = Number of independent variables = 7 Factor 1: % Explained variation = 2.1518/7 = 30.74% Factor 2: % Explained variation = 1.8123/7 = 25.89% Factor 3: % Explained variation = 1.4517/7 = 20.73% Q2.3 Factors 1 & 2 account for 30.74% + 25.89% = 56.63% of variation. Factors 1 & 3 account for 30.74% + 20.73% = 51.47% of variation. Factors 2 & 3 account for 25.89% + 20.73% = 46.62% of variation. Since the sum of % explained variation of Factors 1 and 2 is greatest, this is the most representative bivariate plot of the original variables. Q2.4 % of variation attributed to rotated factor 3: done in Q2.2 Q2.5 Factor 1: Product knowledge Factor 2: Interpersonal skills Factor 3: Service 3
QUESTION 3 Q3.1 METHOD 1: Sum communalities across all variables on Unrotated Factor 1… Eigenvalue1 = 0.2613 + 0.3056 + 0.3993 + 0.7495 +0.862238 = 2.5782 METHOD 2: Square and sum factor loadings across all variables on Unrotated Factor 1… Eigenvalue1 = (-0.5112)2 + (-0.5528) 2 + (-0.6319) 2 + (-0.8657) 2 + (-0.9285) 2 = 2.5782 Q3.2 Total amount of variation is equal to number of variables in the model = 5. So the amount of variation explained by Unrotated Factor 1 is 2.5782/5 = 0.5156 or 51.56%. Q3.3 You are given the eigenvalue for Factor 2 (= 1.5672), so amount of variation explained by the two significant unrotated factors is (2.5782 + 1.5672)/5 = 0.8291 or 82.91%. Q3.4 Here you’re given the communalities “From 2 Factors”... Com munalities (Factor12 - Pizza Restaurants Consum er Assessment Study ) Extraction: Principal com ponents Rotation: Unrotated From 1 From 2 Multiple Variable Factor Factors R-Square DECOR 0.261365 0.872676 0.604196 ATMOSPHR 0.305640 0.874859 0.592956 SERVICE 0.399395 0.586072 0.292001 QUALITY 0.749574 0.898775 0.912763 VALUE 0.862238 0.913068 0.919748
Q3.5 Use method 2 from Q1.1: Factor Loadings (Varimax normalized) (Factor12 - Pizza Restaurants Consumer Assessment Study) Extraction: Principal components (Marked loadings are >.700000) Factor Factor Variable 1 2 DECOR 0.055309 0.932532 ATMOSPHR 0.105031 0.929423 SERVICE 0.763115 -0.061047 QUALITY 0.943281 0.094840 VALUE 0.917893 0.265595 Expl.Var 2.328742 1.816706 Prp.Totl 0.465748 0.363341
For Unrotated factors, proportion explained by factor 1 was 51.56% and proportion explained by factor 2 was 31.35%. A small amount of the variation that was being explained by factor 1 has 4
been reallocated and is now being explained by factor 2. Factor 1 has thus given up some of its explanatory power to Factor 2, with the expectation that this leads to better interpretations. Q3.6 F a c to r L o a d in g s , F a c to r 1 v s . F a c tor 2 R o t a t io n : V a r im a x n o r m a liz e d E x t r a c t io n : P r in c ip a l c o m p o n e n t s 1 .0
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Q3.7 Factor 1: Standard of product offered (SERVICE, QUALITY, VALUE) Factor 2: Ambience of restaurant (DÉCOR, ATMOSPHERE)
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