11

International Conference

The Many Dimension of The Many Dimension of Poverty Poverty

Brasilia, BrazilBrasilia, Brazil – 29-31 August 2005 – 29-31 August 2005

22

Methods of Factor Analysis for Methods of Factor Analysis for Ordinal Categorical Poverty DataOrdinal Categorical Poverty Data

Gisele Kamanou, Ph.D.Gisele Kamanou, Ph.D.

Office of the DirectorOffice of the Director

United Nations Statistics Division (UNSD)United Nations Statistics Division (UNSD)

New YorkNew York

The views expressed here are those of the author and do not necessarily The views expressed here are those of the author and do not necessarily reflect those of UNSDreflect those of UNSD

33

Use of factor analysis technique in Use of factor analysis technique in empirical analysis of Poverty:empirical analysis of Poverty:

The study of multidimensional poverty The study of multidimensional poverty involves the joint analysis of several involves the joint analysis of several variablesvariables

poverty is not directly measurablepoverty is not directly measurable

Factor model posits the existence of one Factor model posits the existence of one or more underlying (theoretical) or more underlying (theoretical) continuous variable(s) that would explain continuous variable(s) that would explain the measures taken on the multiple the measures taken on the multiple variablesvariables

44

Factor Analysis addresses 2 Factor Analysis addresses 2 distinct issues distinct issues

Measurement issue: The latent variables Measurement issue: The latent variables are the “true” measuresare the “true” measures

Data reduction issue: the original Data reduction issue: the original variables can be projected on to a lower variables can be projected on to a lower dimensional space of the latent variablesdimensional space of the latent variables

55

Factor Analysis model: Factor Analysis model:

Factor analysis essentially assumes a Factor analysis essentially assumes a model for the correlation matrixmodel for the correlation matrix

Thus, the original data are assumed Thus, the original data are assumed continuous with positive definite continuous with positive definite correlation matrixcorrelation matrix

Yet, mutidimensional poverty data for Yet, mutidimensional poverty data for most part categorical do not meet these most part categorical do not meet these assumptionsassumptions

66

The Mechanics of Factor Analysis: The Mechanics of Factor Analysis: Few data examples: Few data examples:

Gambia dataset: 25 Variables including: Gambia dataset: 25 Variables including: Age group, Relationship, Gender, Marital Age group, Relationship, Gender, Marital Status, Type of marital union, Household Status, Type of marital union, Household size (hhsize), Urban category, Income size (hhsize), Urban category, Income group, Poverty category, Per capita group, Poverty category, Per capita income category, Per capita Income per income category, Per capita Income per adult equivalent unit (pincaeu), Socio adult equivalent unit (pincaeu), Socio economic groupeconomic group..

Intuitively, one would like to fit a factor Intuitively, one would like to fit a factor model to pincaeu, household size, model to pincaeu, household size, marital status and urban category, socio marital status and urban category, socio economic group and gendereconomic group and gender

77

The Mechanics of Factor Analysis The Mechanics of Factor Analysis (cont.)(cont.)

Only two variables are numeric: pincaeu Only two variables are numeric: pincaeu and hhsize (note that pincaeu is function and hhsize (note that pincaeu is function of hhsize)of hhsize)

A factor model fitted to two variables A factor model fitted to two variables generates a negative degree of freedom generates a negative degree of freedom

( )( ))(

2

1)(

2

1 2 kpkps

88

The Mechanics of Factor Analysis (cont.)The Mechanics of Factor Analysis (cont.)

Sierra Leone dataset: 148 variables used to record non-food expenditures including:Sierra Leone dataset: 148 variables used to record non-food expenditures including:

2 Water Charges2 Water Charges 3 Kerosene and other liquid fuel (incl. Palm Kernel Oil)3 Kerosene and other liquid fuel (incl. Palm Kernel Oil) 4 Gas for Cooking4 Gas for Cooking 5 Charcoal5 Charcoal 6 Firewood and Other Solid fuel6 Firewood and Other Solid fuel ************** 7 Repairs to Clothing7 Repairs to Clothing 8 Repairs to Footwear8 Repairs to Footwear 9 Repairs to Soft Furnishings9 Repairs to Soft Furnishings 10 Repairs to Furniture and Fittings10 Repairs to Furniture and Fittings 11 Repairs to Appliances11 Repairs to Appliances 12 Soap and Washing Powder12 Soap and Washing Powder ****************** 13 Insecticides Disinfectants and Household Cleaners13 Insecticides Disinfectants and Household Cleaners 14 Matches14 Matches 15 Toilet paper15 Toilet paper 16 Light Globes / Bulbs16 Light Globes / Bulbs 17 Candles17 Candles 18 Other Non-durable goods18 Other Non-durable goods 19 Household services (Lawns Boy Washman etc.)19 Household services (Lawns Boy Washman etc.) 20 Cooks20 Cooks 21 Baby Sitters / Day Care Attendants Nfnfies Cleaners21 Baby Sitters / Day Care Attendants Nfnfies Cleaners 22 Gardeners22 Gardeners 23 Security Guards23 Security Guards 24 Washmen24 Washmen 25 Plumbing & Repairs25 Plumbing & Repairs 26 Furniture and Furnishing Hire26 Furniture and Furnishing Hire ************ 27 Pain Killers (Aspirin.)27 Pain Killers (Aspirin.) 28 Antibiotics28 Antibiotics 29 Anti malaria medicines29 Anti malaria medicines 30 Other Medical and Pharmaceutical Products30 Other Medical and Pharmaceutical Products 31 Medical Services such as Doctor or Healer and other Medical31 Medical Services such as Doctor or Healer and other Medical ********** 32 Car32 Car 33 Motorcycle33 Motorcycle 34 Bicycle34 Bicycle 35 Boat35 Boat 36 Other Personal Transport Equipment36 Other Personal Transport Equipment 37 Tyres Tubes Pants37 Tyres Tubes Pants 38 Other Accessories38 Other Accessories 39 Petrol39 Petrol 40 Diesel40 Diesel 41 Oil & Grease41 Oil & Grease 42 Chiffonier and Driver Services42 Chiffonier and Driver Services 43 Other Equipment including repairs & serving43 Other Equipment including repairs & serving 44 Road a rail Transport44 Road a rail Transport 45 Inland water Transport45 Inland water Transport 46 Air Transport46 Air Transport 47 Ocean Transport47 Ocean Transport 48 Other Transport48 Other Transport 49 Postal Charges (Including stamps and courier services)49 Postal Charges (Including stamps and courier services) 50 Telegrams Telephones Faxes etc50 Telegrams Telephones Faxes etc ******************** 51 School Fees and Related Charges51 School Fees and Related Charges 52 School Books and Stationery52 School Books and Stationery 53 School Transport53 School Transport 54 Boarding Lodging at School54 Boarding Lodging at School 55 Other Expenses on Education55 Other Expenses on Education 56 School Uniform56 School Uniform 57 TV Video Radio Cassette Player57 TV Video Radio Cassette Player 58 Musical Instruments58 Musical Instruments 59 Camera Video Camera and Other Durable Photographic Equipmen59 Camera Video Camera and Other Durable Photographic Equipmen 60 Typewriter Binoculars Sports Equipment60 Typewriter Binoculars Sports Equipment 61 Other Accessories and Repairs61 Other Accessories and Repairs 62 Books and Magazines62 Books and Magazines 63 Newspapers63 Newspapers 64 Transistor Batteries / Films & Other Non-Durable Photo Items64 Transistor Batteries / Films & Other Non-Durable Photo Items 65 Other Non-durable Items65 Other Non-durable Items 66 Football Cinema Video Tickets and Charges66 Football Cinema Video Tickets and Charges 67 Membership of Sports Video Societies and Other Clubs67 Membership of Sports Video Societies and Other Clubs 68 Stationery Supplies – Wring Pad Pens Pall Pens Pencils I68 Stationery Supplies – Wring Pad Pens Pall Pens Pencils I 69 Drawing Equipment and Accessories69 Drawing Equipment and Accessories 70 Other Items and Repairs70 Other Items and Repairs 71 Personal Care Services71 Personal Care Services 72 Barber Beauty Saloon72 Barber Beauty Saloon 73 Other Personal Care services73 Other Personal Care services 74 Stationery Supplies – Wring Pad Pens Ball Pens Pencils I74 Stationery Supplies – Wring Pad Pens Ball Pens Pencils I 75 Drawing Equipment and Accessories75 Drawing Equipment and Accessories 76 Other Items and Repairs76 Other Items and Repairs 77 Services of Barber Beauty Shops and Others77 Services of Barber Beauty Shops and Others 78 Personal Care Services78 Personal Care Services 79 Barber Beauty Salon79 Barber Beauty Salon 80 Other Personal Care services80 Other Personal Care services 81 Goods for Personal Care (eg. Razor Blades Cosmetics Powder81 Goods for Personal Care (eg. Razor Blades Cosmetics Powder 82 Writing and Drawing Equipment and Supplies82 Writing and Drawing Equipment and Supplies 83 Expenditure in Restaurants and Hotels83 Expenditure in Restaurants and Hotels 84 Financial Services (N.E.S = Not Elsewhere Specified)84 Financial Services (N.E.S = Not Elsewhere Specified) 85 Jewellery & Watcher85 Jewellery & Watcher 86 Other Personal Goods86 Other Personal Goods 87 Package Tours87 Package Tours 88 Goods N.E.C88 Goods N.E.C 89 Other Services (N.E.S)89 Other Services (N.E.S) **************** 90 Cotton90 Cotton 91 Silk91 Silk 92 Hand Loomed Cloth92 Hand Loomed Cloth 93 Polyester Material93 Polyester Material 94 All Other Clothing Material (Natural fibre or otherwise)94 All Other Clothing Material (Natural fibre or otherwise) 95 Tailoring Charges95 Tailoring Charges 96 Suit / Safari Suit96 Suit / Safari Suit 97 Smock or Other Hand Woven Garment97 Smock or Other Hand Woven Garment 98 Dress (ladies/girls)98 Dress (ladies/girls) 99 Trousers Slacks Shorts Blouse Shirts99 Trousers Slacks Shorts Blouse Shirts 100 Underwear (Incl.Vests and Underpants)100 Underwear (Incl.Vests and Underpants) 101 Other Readymade Clothes101 Other Readymade Clothes 102 Mens Shoes102 Mens Shoes 103 Mens Slippers103 Mens Slippers 104 Ladies Shoes104 Ladies Shoes 105 Ladies Slippers105 Ladies Slippers 106 Boys shoes106 Boys shoes 107 Girls shoes107 Girls shoes 108 Repair of footwear108 Repair of footwear 109 House / Property Rates109 House / Property Rates 110 Basic Rates110 Basic Rates 111 Bed sheets Bed Cover Blanket Curtains111 Bed sheets Bed Cover Blanket Curtains 112 Mattress Pillow & Cases112 Mattress Pillow & Cases 113 Bed / Sleeping Mats113 Bed / Sleeping Mats 114 Chair114 Chair 115 Table115 Table 116 Carpet and other floor covering116 Carpet and other floor covering 117 Cupboards/Wardrobes Repairs117 Cupboards/Wardrobes Repairs 118 Other furniture and fixtures118 Other furniture and fixtures 119 Electric fan119 Electric fan 120 Air Conditioner / Air Cooler120 Air Conditioner / Air Cooler 121 Fridge and Freezers121 Fridge and Freezers 122 Electric Iron122 Electric Iron 123 Washing Machine and dryer123 Washing Machine and dryer 124 Electric Kettle124 Electric Kettle 125 Gas or Electric Stoves125 Gas or Electric Stoves 126 Spectacles and contact Lenses126 Spectacles and contact Lenses 127 Wheelchairs127 Wheelchairs 128 Crutches128 Crutches 129 Other appliances (medical)129 Other appliances (medical) 130 Dentist fees130 Dentist fees 131 Nurse Midwives etc.Fees131 Nurse Midwives etc.Fees 132 Traditional and Spiritual Healers132 Traditional and Spiritual Healers 133 Pharmacist Fees (Consultation)133 Pharmacist Fees (Consultation) 134 Other Medical Practitioners134 Other Medical Practitioners 135 Headache pills135 Headache pills 136 Other medicines136 Other medicines 137 Cars or Motor Vehicle137 Cars or Motor Vehicle 138 Motor Cycles138 Motor Cycles 139 Bicycles139 Bicycles 140 Tyres140 Tyres 141 Radio Wireless and Cassette/Radio141 Radio Wireless and Cassette/Radio 142 T.V. Sets Video Video Camera142 T.V. Sets Video Video Camera 143 Other (phonogram C/D players music systems)143 Other (phonogram C/D players music systems) 144 Camera and photographic equipment144 Camera and photographic equipment 145 Sports Equipments145 Sports Equipments 146 Musical Instruments146 Musical Instruments 147 Jewellery(watches rings etc)147 Jewellery(watches rings etc) 148 Other personal goods148 Other personal goods

99

The Mechanics of Factor Analysis The Mechanics of Factor Analysis (cont.)(cont.)

Exploratory Factor analysis handy in this Exploratory Factor analysis handy in this case with a dual objective:case with a dual objective:

• 1) to address the issue of 1) to address the issue of measurement errors that is likely to be measurement errors that is likely to be present in detailed accounts of present in detailed accounts of household expenditures household expenditures

• 2) to reduce the many expenditure 2) to reduce the many expenditure variables to few factors which account variables to few factors which account for the covariability among the initial for the covariability among the initial expenditure variables. expenditure variables.

3 Illustrative examples:3 Illustrative examples:

)(2

1)(

2

1 2 kpkps

1010

The Mechanics of Factor Analysis (cont.)The Mechanics of Factor Analysis (cont.)

Model1:Model1: The observed (called manisfest) variables include The observed (called manisfest) variables include Water Charges, Kerosene and other liquid Water Charges, Kerosene and other liquid

fuel, Charcoal and Firewood and Other Solid fuelfuel, Charcoal and Firewood and Other Solid fuel..

• fnf.278 fnf.279 fnf.281 fnf.282 fnf.278 fnf.279 fnf.281 fnf.282 • Min.: 1400.0 Min.: 6000 Min.: 2000.00 Min.: 3000 Min.: 1400.0 Min.: 6000 Min.: 2000.00 Min.: 3000 • 1st Qu.: 13000.0 1st Qu.: 55200 1st Qu.: 24000.00 1st Qu.: 56000 1st Qu.: 13000.0 1st Qu.: 55200 1st Qu.: 24000.00 1st Qu.: 56000 • Median: 35000.0 Median: 90000 Median: 60000.00 Median:127400 Median: 35000.0 Median: 90000 Median: 60000.00 Median:127400 • Mean: 77534.7 Mean: 132425 Mean: 94288.89 Mean:158910 Mean: 77534.7 Mean: 132425 Mean: 94288.89 Mean:158910 • 3rd Qu.: 88000.0 3rd Qu.: 180000 3rd Qu.: 120000.00 3rd Qu.:196800 3rd Qu.: 88000.0 3rd Qu.: 180000 3rd Qu.: 120000.00 3rd Qu.:196800 • Max.:920000.0 Max.:1000000 Max.:1200000.00 Max.:728000 Max.:920000.0 Max.:1000000 Max.:1200000.00 Max.:728000

• Importance of factors:Importance of factors:• Factor1 Factor1 • SS loadings 0.34153197SS loadings 0.34153197• Proportion Var 0.08538299Proportion Var 0.08538299• Cumulative Var 0.08538299Cumulative Var 0.08538299

• The degrees of freedom for the model is 2.The degrees of freedom for the model is 2.

• Uniquenesses:Uniquenesses:• fnf.278 fnf.279 fnf.281 fnf.282 fnf.278 fnf.279 fnf.281 fnf.282 • 0.8752831 0.9994841 0.7930973 0.99060350.8752831 0.9994841 0.7930973 0.9906035

• Loadings:Loadings:• Factor1 Factor1 • fnf.278 0.353 fnf.278 0.353 • fnf.279 fnf.279 • fnf.281 0.455 fnf.281 0.455 • fnf.282 fnf.282

• Importance of factors:Importance of factors:• Factor1 Factor2 Factor1 Factor2 • SS loadings 0.37157712 0.19891522SS loadings 0.37157712 0.19891522• Proportion Var 0.09289428 0.04972881Proportion Var 0.09289428 0.04972881• Cumulative Var 0.09289428 0.14262308Cumulative Var 0.09289428 0.14262308

• The degrees of freedom for the model is -1.The degrees of freedom for the model is -1.

• Uniquenesses:Uniquenesses:• fnf.278 fnf.279 fnf.281 fnf.282 fnf.278 fnf.279 fnf.281 fnf.282 • 0.840845 0.8455922 0.7586841 0.98438630.840845 0.8455922 0.7586841 0.9843863• Loadings:Loadings:• Factor1 Factor2 Factor1 Factor2 • fnf.278 0.384 -0.108 fnf.278 0.384 -0.108 • fnf.279 0.393 fnf.279 0.393 • fnf.281 0.462 0.167 fnf.281 0.462 0.167 • fnf.282 -0.102 fnf.282 -0.102

1111

The Mechanics of Factor Analysis (cont.)The Mechanics of Factor Analysis (cont.)

Poor fit: 2 possible explanationsPoor fit: 2 possible explanations

• The manifest variables (The manifest variables (Water Charges, Water Charges, Kerosene and other liquid fuel, Charcoal Kerosene and other liquid fuel, Charcoal and Firewood and Other Solid fueland Firewood and Other Solid fuel) have ) have very low correlationvery low correlation

• The data do not meet the model The data do not meet the model assumption (the most obvious)assumption (the most obvious)

1212

The Mechanics of Factor Analysis (cont.)The Mechanics of Factor Analysis (cont.)

Model2 with 4 manisfest variables: Model2 with 4 manisfest variables: Pain Pain Killers (Aspirin), Tailoring Charges, Killers (Aspirin), Tailoring Charges, Underwear and Ladies SlippersUnderwear and Ladies Slippers. .

Poor fit statistics – but some evidence thatPoor fit statistics – but some evidence that

• one factor model is more adequate than one factor model is more adequate than a two factor model (negative df)a two factor model (negative df)

• the variable fnf.303 (Pain Killers) has low the variable fnf.303 (Pain Killers) has low covariance with the other three covariance with the other three variables. variables.

1313

The Mechanics of Factor Analysis (cont.)The Mechanics of Factor Analysis (cont.)Model 2:Model 2:

[1] 2456 148[1] 2456 148

fnf.303 inf.206 inf.211 inf.216 fnf.303 inf.206 inf.211 inf.216 Min.: 16.0 Min.: 9.0 Min.: 15.0 Min.: 12.0 Min.: 16.0 Min.: 9.0 Min.: 15.0 Min.: 12.0 1st Qu.: 5600.0 1st Qu.: 8000.0 1st Qu.: 5000.0 1st Qu.: 6000.0 1st Qu.: 5600.0 1st Qu.: 8000.0 1st Qu.: 5000.0 1st Qu.: 6000.0 Median: 12000.0 Median: 15000.0 Median: 10000.0 Median: 12000.0 Median: 12000.0 Median: 15000.0 Median: 10000.0 Median: 12000.0 Mean: 24758.9 Mean: 26132.3 Mean: 18448.8 Mean: 16929.9 Mean: 24758.9 Mean: 26132.3 Mean: 18448.8 Mean: 16929.9 3rd Qu.: 29600.0 3rd Qu.: 28000.0 3rd Qu.: 20000.0 3rd Qu.: 20000.0 3rd Qu.: 29600.0 3rd Qu.: 28000.0 3rd Qu.: 20000.0 3rd Qu.: 20000.0 Max.:400000.0 Max.:540000.0 Max.:600000.0 Max.:224000.0 Max.:400000.0 Max.:540000.0 Max.:600000.0 Max.:224000.0

Importance of factors:Importance of factors: Factor1 Factor1 SS loadings 1.3698167SS loadings 1.3698167 Proportion Var 0.3424542Proportion Var 0.3424542 Cumulative Var 0.3424542Cumulative Var 0.3424542

The degrees of freedom for the model is 2.The degrees of freedom for the model is 2.

Uniquenesses:Uniquenesses: fnf.303 inf.206 inf.211 inf.216 fnf.303 inf.206 inf.211 inf.216 0.9647335 0.4783844 0.6729641 0.51410130.9647335 0.4783844 0.6729641 0.5141013

Loadings:Loadings: Factor1 Factor1 fnf.303 0.188 fnf.303 0.188 inf.206 0.722 inf.206 0.722 inf.211 0.572 inf.211 0.572 inf.216 0.697 inf.216 0.697

Importance of factors:Importance of factors: Factor1 Factor2 Factor1 Factor2 SS loadings 1.2583571 0.2359544SS loadings 1.2583571 0.2359544 Proportion Var 0.3145893 0.0589886Proportion Var 0.3145893 0.0589886 Cumulative Var 0.3145893 0.3735779Cumulative Var 0.3145893 0.3735779

The degrees of freedom for the model is -1.The degrees of freedom for the model is -1.

Uniquenesses:Uniquenesses: fnf.303 inf.206 inf.211 inf.216 fnf.303 inf.206 inf.211 inf.216 0.8588318 0.4875558 0.6747178 0.48458320.8588318 0.4875558 0.6747178 0.4845832 Loadings:Loadings: Factor1 Factor2 Factor1 Factor2 fnf.303 0.364 fnf.303 0.364 inf.206 0.680 0.222 inf.206 0.680 0.222 inf.211 0.532 0.206 inf.211 0.532 0.206 inf.216 0.710 0.108 inf.216 0.710 0.108

1414

The Mechanics of Factor Analysis (cont.)The Mechanics of Factor Analysis (cont.)

Model3 with 5 manisfest variables: Model3 with 5 manisfest variables: Insecticide Insecticide Disinfectants and Household Cleaners, Matches, Disinfectants and Household Cleaners, Matches, School Books and Stationery, Other Expenses on School Books and Stationery, Other Expenses on Education and, School Uniform Education and, School Uniform

• Better statistics compared to the Model1 and Better statistics compared to the Model1 and Model2 Model2

• We reasonably posit the existence of two factors We reasonably posit the existence of two factors which jointly account for nearly 53% of the total which jointly account for nearly 53% of the total variance in the original variablesvariance in the original variables

• A straight forward interpretation: factor1 A straight forward interpretation: factor1 represents school related expenditures and represents school related expenditures and factor2 represents the expenditures on factor2 represents the expenditures on houseware comestibles (that is, matches and houseware comestibles (that is, matches and insecticides) insecticides)

1515

The Mechanics of Factor Analysis (cont.)The Mechanics of Factor Analysis (cont.)Model 3Model 3

• Importance of factors:Importance of factors:• Factor1 Factor1 • SS loadings 2.112810SS loadings 2.112810• Proportion Var 0.422562Proportion Var 0.422562• Cumulative Var 0.422562Cumulative Var 0.422562

• The degrees of freedom for the model is 5.The degrees of freedom for the model is 5.

• Uniquenesses:Uniquenesses:• fnf.289 fnf.290 fnf.328 fnf.331 fnf.332 fnf.289 fnf.290 fnf.328 fnf.331 fnf.332 • 0.9929941 0.9965059 0.2016838 0.4186203 0.2773860.9929941 0.9965059 0.2016838 0.4186203 0.277386

• Loadings:Loadings:• Factor1 Factor1 • fnf.289 fnf.289 • fnf.290 fnf.290 • fnf.328 0.893 fnf.328 0.893 • fnf.331 0.762 fnf.331 0.762 • fnf.332 0.850 fnf.332 0.850

• Importance of factors:Importance of factors:• Factor1 Factor2 Factor1 Factor2 • SS loadings 2.1228869 0.5142867SS loadings 2.1228869 0.5142867• Proportion Var 0.4245774 0.1028573Proportion Var 0.4245774 0.1028573• Cumulative Var 0.4245774 0.5274347Cumulative Var 0.4245774 0.5274347

• The degrees of freedom for the model is 1.The degrees of freedom for the model is 1.

• Uniquenesses:Uniquenesses:• fnf.289 fnf.290 fnf.328 fnf.331 fnf.332 fnf.289 fnf.290 fnf.328 fnf.331 fnf.332 • 0.76664 0.779216 0.2244735 0.3869945 0.20550240.76664 0.779216 0.2244735 0.3869945 0.2055024• Loadings:Loadings:• Factor1 Factor2 Factor1 Factor2 • fnf.289 0.481 fnf.289 0.481 • fnf.290 0.469 fnf.290 0.469 • fnf.328 0.878 fnf.328 0.878 • fnf.331 0.751 0.221 fnf.331 0.751 0.221 • fnf.332 0.886 fnf.332 0.886

1616

Principal Component analysis vs. factor Principal Component analysis vs. factor analysis analysis

• Both obtain a reduced-rank representation of Both obtain a reduced-rank representation of a set of observed variablesa set of observed variables with a with a minimal minimal loss of informationloss of information

• PCA aims at retaining the maximum variance PCA aims at retaining the maximum variance in the original data whereas a factor model in the original data whereas a factor model attempts to fully account for the attempts to fully account for the multicolinearity of the original variablesmulticolinearity of the original variables

1717

Principal Component analysis vs factor Principal Component analysis vs factor analysis (cont)analysis (cont)

Further:Further:

• PCA is a purely analytical tool which makes no PCA is a purely analytical tool which makes no prior supposition on the structure of the data prior supposition on the structure of the data or on the relationship among the variablesor on the relationship among the variables

• In contrast to PCA, factor analysis was In contrast to PCA, factor analysis was developed to address a measurement issue developed to address a measurement issue ((Journal of Consumer PsychologyJournal of Consumer Psychology 2001) under 2001) under the basic assumption that the co-variability the basic assumption that the co-variability that is common to all measured variables is that is common to all measured variables is attributed to the underlying latent variables, attributed to the underlying latent variables, also called common factors. also called common factors.

1818

Principal component analysis vs factor Principal component analysis vs factor analysis (cont)analysis (cont)

Algebraically:Algebraically:

• PCA solution is the linear transformationPCA solution is the linear transformation

such that the residual dispersion matrix is such that the residual dispersion matrix is minimumminimum

(min )(min )

• The solution is given by the Single Value The solution is given by the Single Value Decomposition of ,which is assumed to be Decomposition of ,which is assumed to be semi-positive definitesemi-positive definite

XTY '

'1' )( TTTT

1919

Principal Component analysis vs factor Principal Component analysis vs factor analysis (cont)analysis (cont)

Algebraically:Algebraically:

• Factor model also posits a linear formulationFactor model also posits a linear formulation

and a factor model solution will approximate and a factor model solution will approximate

byby

That is, (1)That is, (1)

• The measure of closeness between original The measure of closeness between original data and its approximation is chosen to be the data and its approximation is chosen to be the amount of covariance in the original data amount of covariance in the original data explained by the factors explained by the factors

X

...

...

221

2121111

FaFaX

FaFaX

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q FAX

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2020

Estimation methods in Factor analysisEstimation methods in Factor analysis

• The general formulation of the factor model is The general formulation of the factor model is to approximate the covariance matrix in to approximate the covariance matrix in which a diagonal matrix of estimated unique which a diagonal matrix of estimated unique variances is subtracted by a matrix of variances is subtracted by a matrix of reproduced correlations, that is: reproduced correlations, that is:

• The optimum choice for will be such that The optimum choice for will be such that the off diagonal elements of correlation the off diagonal elements of correlation matrix of the residuals ( ) are as small matrix of the residuals ( ) are as small as possible as possible

*'ˆfAA

*f

*f

2121

Estimation methods in Factor analysis Estimation methods in Factor analysis (cont)(cont)

• The estimates of the coefficients are not The estimates of the coefficients are not trivial strong assumptions has to be made in trivial strong assumptions has to be made in practice practice

• Several procedures are used in practice to fit Several procedures are used in practice to fit to , and to , and

including the maximum likelihood (ML), the including the maximum likelihood (ML), the unweighted least squares (ULS) and the unweighted least squares (ULS) and the generalized least squares (GLS) techniques. generalized least squares (GLS) techniques.

'AA

A

2222

Estimation methods in Factor analysis Estimation methods in Factor analysis (cont)(cont)

• All these estimation techniques are based on All these estimation techniques are based on a normality assumption of one sort or a normality assumption of one sort or another. another.

• In the ML case for example, it is generally In the ML case for example, it is generally assumed that the residuals in (1) are normal assumed that the residuals in (1) are normal distributed, i.e.distributed, i.e.

~ ~

and that ~ and that ~

such that ~ such that ~

),0( MVN

)(qF ),0( MVN

X ),0( MVN

2323

ML estimation techniques in factor ML estimation techniques in factor analysisanalysis

• Most commonly used methodMost commonly used method

• default method used in the majority of default method used in the majority of statistical packagestatistical package

• It uses the Pearson standard product-moment It uses the Pearson standard product-moment correlation. correlation.

• The Pearson correlation, however, presents a The Pearson correlation, however, presents a number of limits when the data do not meet number of limits when the data do not meet the normality distributional assumptions (e.g. the normality distributional assumptions (e.g. Babakus E., Ferguson C and Joreskog G,1987) Babakus E., Ferguson C and Joreskog G,1987)

2424

Alternative correlation matrix of Alternative correlation matrix of categorical ordinal datacategorical ordinal data

• Bartelemew 1980, develops an iterative Bartelemew 1980, develops an iterative approach to estimate factor scores when the approach to estimate factor scores when the observed variables are dichotomousobserved variables are dichotomous

• Alternative estimation methods for the a Alternative estimation methods for the a common factor model of dichotomous data common factor model of dichotomous data are discussed in Robert Mislevy (1986) are discussed in Robert Mislevy (1986) including the Unweighted Least-Squares including the Unweighted Least-Squares methods, the Generalized Least Squares methods, the Generalized Least Squares solutions and the Maximum Likelihood solutions and the Maximum Likelihood solutionssolutions

2525

Alternative Correlation matrix in factor Alternative Correlation matrix in factor analysis of categorical ordinal data (cont)analysis of categorical ordinal data (cont)

• Several arguments against the Pearson Several arguments against the Pearson correlation in factor analysis of discrete ordinal correlation in factor analysis of discrete ordinal data in the general case: data in the general case:

Discrete and ordinal data do not necessarily produce Discrete and ordinal data do not necessarily produce semi-positive definite correlation matricessemi-positive definite correlation matrices

Factor analysis parameter estimates based on the Factor analysis parameter estimates based on the Pearson’s correlation are biased and model fit severely Pearson’s correlation are biased and model fit severely distorted (Johnson and Creech 1983 and others)distorted (Johnson and Creech 1983 and others)

Dichotomous variables are bounded, implying that Dichotomous variables are bounded, implying that their regression on any continuous latent variable with their regression on any continuous latent variable with finite range cannot be linearfinite range cannot be linear

The linear factor model applied to directly to The linear factor model applied to directly to correlations from dichotomous variables will thus be correlations from dichotomous variables will thus be mis-specified. mis-specified.

2626

Correlation of a contingency table and Correlation of a contingency table and application to the analysis of application to the analysis of multidimensional povertymultidimensional poverty

• In many situations in multidimensional poverty, In many situations in multidimensional poverty, and for non-monetary variables in particular, and for non-monetary variables in particular, variables are categorical or ordinal and often variables are categorical or ordinal and often take values within a small range of discrete take values within a small range of discrete categoriescategories

• In these situations, the contingency table of the In these situations, the contingency table of the variables is used in lieu of the correlation matrixvariables is used in lieu of the correlation matrix

• The polychoric correlation introduced by The polychoric correlation introduced by Ritchie-Scott (1918) and Pearson and Pearson Ritchie-Scott (1918) and Pearson and Pearson (1922), is an alternative to the Pearson (1922), is an alternative to the Pearson correlation specifically for situation where the correlation specifically for situation where the variables are continuous but the measurement variables are continuous but the measurement instruments yield data that may only be ordinalinstruments yield data that may only be ordinal

2727

Correlation of a contingency table and Correlation of a contingency table and application to the analysis of multidimensional application to the analysis of multidimensional poverty (cont)poverty (cont)

• It has been shown that the polychoric correlation It has been shown that the polychoric correlation coefficient, calculated from ordinal transformation coefficient, calculated from ordinal transformation of bivariate normal variables, is an unbiaised of bivariate normal variables, is an unbiaised estimate of the correlation between the original estimate of the correlation between the original bivariate variables (Rigdon and Ferguson 1991, bivariate variables (Rigdon and Ferguson 1991, pp491) pp491)

• It is a better measure of correlation for ML factor It is a better measure of correlation for ML factor analysis of ordinal data (Rigdon and Ferguson, analysis of ordinal data (Rigdon and Ferguson, 1991; Joreskog and Sorbom,1981)1991; Joreskog and Sorbom,1981)

• This option has been implemented in some This option has been implemented in some computer programmes used in the field of computer programmes used in the field of psychology and education (e.g. PRELIS, LISREL)psychology and education (e.g. PRELIS, LISREL)

• but it remains to be implemented in commonly but it remains to be implemented in commonly used statistical packages used social researchused statistical packages used social research

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Concluding notesConcluding notes

• Factor analysis is an important exploratory Factor analysis is an important exploratory analytical tool for quantitative poverty analytical tool for quantitative poverty measures measures

• Factor analysis modeling remains however Factor analysis modeling remains however under developed for qualitative analysis of under developed for qualitative analysis of poverty and in particular for poverty variables poverty and in particular for poverty variables that are categoricalthat are categorical

• The aim of this paper was to raise cautions The aim of this paper was to raise cautions when applying factor analysis mechanically to when applying factor analysis mechanically to data that are not continuous such as those data that are not continuous such as those used to capture the multidimensional aspects used to capture the multidimensional aspects of poverty of poverty

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Concluding notes (cont)Concluding notes (cont)

• Recent developments on factor analysis for Recent developments on factor analysis for categorical data are promising and methods categorical data are promising and methods for factor analysis based on alternative for factor analysis based on alternative estimates of correlation matrix have been estimates of correlation matrix have been proposedproposed

• These methods remain to be implemented in These methods remain to be implemented in the commonly used statistical packages such the commonly used statistical packages such as S-plusas S-plus

• It is hoped that future empirical research will It is hoped that future empirical research will devote due effort to addressing this devote due effort to addressing this shortcoming.shortcoming.