Factor_analysis in psychology clinical biostatistics

Factor analysis

1. FACTOR ANALYSIS By Dr. C. B. Tripathi Department of Biostatistics Institute of Human Behavior and Allied Sciences (IHBAS) Dilshad Garden, Delhi-95, INDIA. E- mali : [email protected] [email protected]

2. FACTOR ANALYSIS: Factor analysis is a statistical tool for analyzing scores on large numbers of variables to determine whether there are any identifiable dimensions that can be used to describe many of the variables under study. The purpose of factor analysis is to discover simple patterns in the pattern of relationships among the variables. In particular, it seeks to discover if the observed variables can be explained largely or entirely in terms of a much smaller number of variables called factor.

3. Example: Consider various measures of activity of the autonomic nervous system-heart rat, blood pressure, etc. Psychologists have wanted to know whether, except for random fluctuation, all those measure move up and down together the “activation” hypothesis or do groups of autonomic measures move up and down together, but separate from other groups or are all the measures largely independent.

4. Example: Factor analysis was invented nearly 100 years ago by psychologist Charles Spearman, who hypothesized that the enormous variety of tests of mental ability: measures of mathematical skill, vocabulary , other verbal skills artistic skills, logical reasoning ability, etc. could all be explained by one underlying ‘factor’ of general intelligence that he called g.

5. He hypothesized that if g could be measured and you could select a subpopulation of people with the same score on g, in that subpopulation you would find no correlations among any tests of mental ability. In other words, he hypothesized that g was the only factor common to all . THE GOALS OF FACTOR ANALYSIS: A typical factor analysis suggests to four major questions: 1. How many different factors are needed to explain the pattern of relationships among these variables?

6. 2. What is the nature of those factors? 3. How well do the hypothesized factors explain the observed data? 4. How much purely random or unique variance does each observed variable ? RESEARCH QUESTION: To explicate the types of research question that can be answered using factor analysis, let us consider hypothetical example of a factor analysis situation.

7. Suppose a researcher measures six variables within a sample of adult female participants in a health maintenance organization. Three of these variables are height, arm length, and leg length, aspects of body size. And three are derived from a health history in which subject is asked to report the number of specific episodes occurring in last year. These variables are no. of sore throats, no. of headaches, and no. of earaches.

8. A researcher may want to see: 1. How these variables group-which ones go together and which ones do not. 2. How strongly does each variable go with its group. 3. All together, how many dimensions are needed to explain relationships among the variables.

9. Let us calculate the correlation among theses six considered variables and it is represented as follows: Height Arm Length Leg Length No. of Sore Throats No. of Headaches No. of Earaches Height - High High Low Low Low Arm Length - High Low Low Low Leg Length - Low Low Low No. of Sore Throats High High No. of Headaches - High No. of Earaches -

10. From the above correlation matrix, it can be observed that the three size variables have high inter-correlations And that the three history variables also have high inter-correlations; that is, a women with longer than average legs may have longer than average arms. If such a matrix is factor analyzed a factor matrix defining the two groups of variables, would be derived, as given next slide:

11. Each column in this table reflects one of the variable grouping or factors. The size variables have high values in one column, and the history variables have high values in the other column. This table, indicating the presence of two distinct groups of variables (two factors). This table is able to summarize the information contained in the previous larger correlation matrix. Matrix becomes the 6×2 from 6×6. Variables Factors I II Height High Low Arm length High Low Leg length High Low No. of Sore Throats Low High No. of Headaches Low High No. of Earaches Low High

12. Factor Analysis is very useful for large number of variable (20X20) and when the variables are appeared in random order, rather than neatly arrange according to grouping.  Actually, Factor analysis is a tool through which we may uncover groupings of variables that are not obvious.

13. FACTOR MATRIX: Based on correlation matrix, factor matrix is calculated. In this matrix each row represents one variable included in the factor analysis and each column representing one factor. Each elements within the matrix are the factor loading ranging from -1 to +1 (Which are like correlation of the variable with the factor). I II III h 2 Variables 1 0.85 0.22 0.03 0.77 2 0.15 - - - 3 0.51 - - - 4 0.83 - - - 5 0.26 - - - Eigen values 1.76 % of variance 0.35

14. The square of a factor loading represents the proportion of variance that the item and factor have in common. In other words, this is the proportion of item variance explained by the factor. In matrix on previous slide, first variable (1) has loading of 0.85 on factor I; approximately (0.85) 2 = 0.7225=72% of variance is accounted for by this loading.  77% of item variance is explained by three factors. The eigen value represent the total amount of variance explained by a factor.

15. The average of the squared loadings in a column is obtained by dividing the eigen value by the number of items in the column ( eigen value/n) This average represents the percentage of inter item variance accounted for by the factor. In above example the eigen value is calculated as follow; 0.85 2 +0.15 2 +0.51 2 +0.83 2 +0.26 2 =1.76.  This eigen value of 1.76 is divided by 5 (because there are five variables), yielding 0.352 thus, approximately 35% of total item variance is accounted for by the first factor. Adding the percentage of variance accounted for by each factor tells us how much variance is explained by all the factors.

16. FEATURES OF A SUCCESSFUL FACTOR ANALYSIS: The success of Factor Analysis is assessed on the basis of the following considerations: One of the steps in factor analysis is breaking down the total variation among variables into variations accounted by different factors. The analysis is considered successful when few factors are able to account for a large part of the total variation, say more than 70%.

17. A factor analysis is considered successful when it is really able to find combinations such that some factors have very high loadings (± 1) and the others very low loadings (close to 0) in different variables. The factors should largely be distinct or non - overlapping. A factor analysis is considered successful when the number of identified factors is very small relative to the number of variables. A basic requirement to achieve successful factor analysis is that the underlying common factors are really present.

18. Reference Indrayan A: Medical Biostatistics (2 nd edition) , Chapman & Hall/CRC, Taulor & Francis Group (2008); ISBN- 13: 978-1-58488-887-1 (Hardcover). Sundaram KR, Dwivedi SN, Sreenivas V: Medical Statistics – Principles & Methods , BI Publications Pvt. Ltd (2010); ISBN- 978-81-7225-319-6. Campbell MJ, Machin D: Medical Statistics-A Commonsense Approach (3 rd edition) , John Wiley & Sons, Ltd (1999); ISBN- 0-471-98721-2. Indrayan A: Basic Methods of Medical Research(2 nd edition) , A.I.T.B.S. Publishers (2008); ISBN- 978-81-7473-335-3.

19. To consult a Statistician after an experiment is finished is often merely asking him to conduct a post mortem examination. He can perhaps say what the experiment died of. -------- R. A. Fisher (1938) A well designed study, poorly analyzed, can be rescued by a reanalysis but poorly designed study is beyond the redemption of even sophisticated statistics. ---------David Machin (1999)

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