Research Methodology: Tools

MSc Business Administration Research Methodology: Tools Applied Data Analysis (with SPSS) Lecture 02: Item Analysis / Scale Analysis / Factor Analysis February 2014 Prof. Dr. Jürg Schwarz Lic. phil. Heidi Bruderer Enzler Contents Slide 2 Aims of the Lecture 3 Typical Syntax 4 Introduction 5 An Example of a Construct... 5 What are Constructs?... 7 Evaluating the Quality of an Instrument... 8 Part I: Item Analysis and Scale Analysis 9 Descriptive Analysis... 9 Item Difficulty (Also Called p-value)... 11 Item Discrimination... 13 Reliability Analysis... 15 Part I: Item Analysis and Scale Analysis with SPSS 16 Part II: Factor Analysis 19 Main Steps for Performing a Factor Analysis... 19 Problematic Aspects... 20 Part II: Factor Analysis with SPSS 21 First Step: Select the Variables... 21 Third Step: Determining the Number of Factors... 32 Fifth Step (Optional): Calculate Sum Scales or Factor Scores... 41 References... 42 Appendix 43

Aims of the Lecture Slide 3 You will understand the term "construct". You will understand item difficulty and how this can be calculated with SPSS. You will understand item discrimination and how this can be calculated with SPSS. You will understand Cronbach's Alpha as a measure for reliability, and how it can be calculated with SPSS. You will understand the concept of factor analysis and know how to perform it with SPSS (Principal Component Analysis). Specifically, you will understand how C a correlation matrix is interpreted. determine the "correct" number of factors (Scree Plot, Kaiser criterion). Varimax rotation is used to be better able to interpret a factor solution. to interpret factors in relationship to their meaning. how factor scores and sum scales are calculated. Typical Syntax Slide 4 Reliability analysis RELIABILITY /VARIABLES=ghqconc ghqsleep ghquse ghqdecis /SCALE('ALL VARIABLES') ALL /MODEL=ALPHA /STATISTICS=DESCRIPTIVE /SUMMARY=TOTAL. Items (list incomplete) Cronbachs Alpha Factor analysis FACTOR /VARIABLES ghqconc ghqconfi ghqdecis ghqenjoy Items (list incomplete) /MISSING LISTWISE /ANALYSIS ghqconc ghqconfi ghqdecis ghqenjoy /PRINT INITIAL CORRELATION SIG KMO INV EXTRACTION ROTATION /PLOT EIGEN ROTATION /CRITERIA MINEIGEN(1) ITERATE(25) Extractions method /EXTRACTION PC /CRITERIA ITERATE(25) Rotation /ROTATION VARIMAX /SAVE REG(ALL) /METHOD=CORRELATION. Saving factor scores

Introduction An Example of a Construct Measuring psychosocial well-being with the "General Health Questionnaire"? In the 2003 Health Survey for England the "General Health Questionnaire" (GHQ-12) was used to measure "psychosocial well-being". The GHQ-12 is a battery of 12 items. Slide 5 An extract from the questionnaire: Question I: Are these 12 items collectively suitable for measuring a construct? => Perform an item analysis and a scale analysis Underlying dimensions of the General Health Questionnaire? There could be groups of items on the GHQ-12 that measure different aspects (dimension, factor) of "psychosocial well-being" Slide 6 ghqdecis Felt capable of making decisions ghquse Felt playing useful part in things ghqface Been able to face problems ghqenjoy Able to enjoy day-to-day activities Positive health Construct "psychosocial well-being" ghqconc Able to concentrate ghqhappy Been feeling reasonably happy ghqunhap Been feeling unhappy and depressed ghqstrai Felt constantly under strain Dimensions (factors) of "psychosocial well-being" ghqconfi Been losing confidence in self ghqsleep Lost sleep over worry Psychological distress ghqover Felt couldn t overcome difficulties ghqworth Been thinking of self as worthless Question II: Does the GHQ-12 have a structure? Are there dimensions that must be considered as separate? => Perform a factor analysis

What are Constructs? A construct is an abstract idea that cannot be directly observed and measured. Factor technical term Slide 7 Dimension theoretical term Item in the questionnaire Indicator theoretical term Examples: Psychosocial well-being, motivation, anxiety, employee satisfaction Complex constructs often contain various aspects (dimensions, factors). Examples: Psychosocial well-being = positive health + psychological distress Stress reactivity = Play-dead reflex + heart phobia + negativism + disturbance Intelligence = Verbal intelligence + mathematical-logical intelligence + C In order to measure constructs, indicators that are themselves measurable must be found. Example: Social status = Income + years of education + job category In order to measure constructs, multiple items with rating scales are used often. It is assumed that these items are an indicator for the construct. Technically, a construct is therefore often regarded as an item battery. In SPSS, articles and this script, an item battery is also designated as a scale. Example: General Health Questionnaire (GHQ-12) There are different possibilities for compressing these items into a score. As a sum scale: The item values of an item battery are summed up. An example is the Apgar Score (determining the vital signs of a newborn). Evaluating the Quality of an Instrument Slide 8 How can the quality of an instrument be assessed? Common sense: Content considerations Item analysis und scale analysis Descriptive analysis Item difficulty Item discrimination Reliability analysis Examination of dimensionality Factor analysis "Part I" of the lecture "Part II" of the lecture When do we want to assess the quality of an instrument for which ends? Pretests: Finding out where and how a questionnaire can be improved. Existing datasets: Choosing appropriate items for further analysis.

Part I: Item Analysis and Scale Analysis Descriptive Analysis Example GHQ-12 in the Health Survey for England 2003 (n = 8,833) Slide 9 Missing values: Around 6% missing values appears unproblematic. In addition, all items have a similar proportion of missing values. ghquse ("felt playing useful part in things") may be somewhat vague, and so less frequently answered. Mean / Median / Skewness / Kurtosis: Difference between mean and median distributions are not symmetric. All distributions are more or less right skewed (skewness > 0) and more peaked than a normal distribution (kurtosis > 0). Standard deviation: Variability is assessed with the help of the histograms (see below). Minimum / Maximum: Values between 1 and 4 The entire range of the scaled was used. Example GHQ-12: Histograms Slide 10 ghqconc ghqsleep ghquse ghqdecis ghqstrai ghqover ghqenjoy ghqface ghqunhap ghqconfi ghqworth ghqhappy Variables ghqconc, ghquse, ghqdecis, ghqenjoy, ghqface and ghqhappy: Items display little variability, could be problematic. The most frequently chosen category 2 represents "same as always". Variables qhqsleep, ghqstrai, ghqover, qhqunhap, ghqconfi and ghqworth: These items exhibit more variance, but are strongly skewed to the right. Floor effects, especially with qhqunhap, ghqconfi and ghqworth. The most commonly chosen value of 1 represents an especially good condition of health.

Item Difficulty (Also Called p-value) The difficulty of an item indicates the proportion of respondents who respond to the item in a manner that indicates the characteristic in question is present to a greater extent. Slide 11 Computing item difficulty Continuous items (Rating scales): Item difficulty = arithmetic mean of the item Binary items (for example, Yes/No): Item difficulty is the proportion of respondents who answered a question affirmatively or correctly, when the coding is: affirmative or correct = 1, negative or false = 0 Item difficulty Item i Number of correct responsesitem i = p= Number of valid responses Item i Interpretation of item difficulty High value Item is "easy" Low value Item is "difficult" As a rule, a good mix in an item battery is desirable. Items with extreme values should be revised. With binary items the p-value lies between 0 and 1 (.5 = middle difficulty): Items with p <.1 (very difficult) or p >.9 (very easy) should be revised. Example item difficulty with binary items: A fictitious questionnaire about taxes For example item 5: "In your opinion, should taxes be lowered? Yes or No?" Slide 12 Respondent Item 1 2 3 4 5 1 1 n 1 1 1 2 1 n 0 0 1 3 1 0 1 0 1 4 1 0 0 0 1 5 1 0 1 0 0 6 1 0 0 0 n 7 1 0 1 1 n ny 7 0 4 2 4 na 7 5 7 7 5 nt 7 7 7 7 7 p 1.00 0.00 0.57 0.29 0.80 Legend 0 = no, 1 = yes, n = no answer ny = number of those who answered "Yes" na = number of those who answered nt = total number of those asked p = item difficulty = ny/na Item 1 is too "easy", no distinction is visible. Item 2 appears to be too "difficult" or is not understood. Both items should be eliminated. Item 3 has an average difficulty. Item 4 is relatively difficult, item 5 relatively simple. An example of items with a rating scale follows below.

Item Discrimination Item discrimination is the correlation between an item and the item battery without this item. Item discrimination is computed for every item. Slide 13 The discrimination value of an item indicates how well this single item predicts the value of the item battery. Values range from -1 to 1. The higher the value, the better the item measures what the item battery measures. Positive values near 1 are desirable. If the value is negative, this may be due to the rotation of the item (direction/polarity). If the rotation is correct, however, the item should be discarded or revised. Rule of thumb: Items with a discrimination value under.30 are discarded or revised. Changing item rotation in SPSS Example: item v01 with for possible values 1, 2, 3, and 4. RECODE v01 (1 = 4)(2 = 3)(3 = 2)(4 = 1) INTO v01_r. FREQUENCIES v01 v01_r. Example GHQ-12 AnalyzeScaleReliability AnalysisC Statistics: "Scale if item deleted" Slide 14 Column "Corrected Item-Total Correlation": Item discrimination ranges from.487 to.738. Thus all values are above the threshold of.30. This means that each of the items reflects sufficiently well what the scale as a whole measures.

Reliability Analysis Reliability analysis is used to quantify how well all items in a battery collectively measure the same theoretical construct, for example, psychosocial well-being. A high average correlation between the items indicates that they all measure the same construct. Slide 15 Usually, reliability is measured through the Cronbach's alpha coefficient. Cronbach's alpha is a measurement of the "internal consistency" of a scale. Cronbach's alpha is a positive function of the average correlation among items in the battery, and is positively correlated with the number of items and the size of the sample. Preconditions for calculating Cronbach's alpha All items measure the same theoretical construct Metric variables All items are scaled in the same direction (for example, low values for the items represent high psychosocial well-being) All items have the same distribution (ideally: normal distribution) Evaluation of Cronbach's alpha Alpha should be >.80. In practice, however, values of.60 or.70 are acceptable. Part I: Item Analysis and Scale Analysis with SPSS General Health Questionnaire (GHQ-12) in the Health Survey for England 2003 (n = 8,833) Slide 16 AnalyzeScaleReliability AnalysisC

Slide 17 Results Case Processing Summary 7.3% of the cases are excluded because of missing values. OK Reliability Statistics The reliability of the scale is high (Cronbach's alpha =.891). OK Item Statistics Since the item difficulty (p-value) is the arithmetic mean, it is found in the "Mean" column. Scale from 1 to 4 average difficulty of 2.5. However, in this case the wording suggests that 2 is a type of middle category ("same as usual"). The item difficulties vary between 1.40 and 2.14. That means that the mixture is not very high; some items are very "difficult" (low p-values). Item-Total Statistics Slide 18 Since item discrimination is the correlation between a single item and the total score without this item, it appears in the column "Corrected Item- Total Correlation". Item discrimination varies between.487 and.738 and therefore lies above.30 OK In the column "Cronbach's Alpha if Item Deleted", we see for each item how high Cronbach's alpha would be if this item were omitted. (Including all items: Alpha =.891) In the example, Cronbach's alpha would not be higher if any of items were omitted. This indicates that all items should remain. OK Summary Question I Each of the items represents the entire battery. However, the total battery displays little variation. From the standpoint of reliability, the GHQ-12 items can be used as a scale.

Part II: Factor Analysis Main Steps for Performing a Factor Analysis 1. Select the variables Include only theoretically relevant variables Sufficient number of variables (4 or more per factor) Not too small a sample Perform descriptive analysis (as in Part I: item analysis and reliability analysis) Consider the correlation matrix Slide 19 2. Test suitability and extract the factors Determination of suitability: Evaluate inverse of correlation matrix, Bartlett's test and KMO Select extraction method (principal component analysis, principal axis analysis) 3. Specify the number of factors Criteria: Eigenvalue, scree plot, rules of thumb 4. Interpret the factors Rotation of the factor matrix, mapping variables to factors, interpretation 5. Optional: Compute sum scales or factor scores Computing sum scales or factor scores Problematic Aspects Slide 20 Many decisions regarding extraction and interpretation of factors are subjective. The same dataset can produce different results, depending on the "decision path". Although variables must be at least interval-scaled, in practice, variables with lower scale levels are often included, which can lead to false conclusions. Sample size There is no scientifically exact rule how large the sample should be. One of the possible rules of thumb suggests that there should be at least 10 subjects per item ("Rule of 10"). Problematic missing values In item batteries, there are often many missing values. Results are different, depending on "missing treatment" in relation to Number of factors Interpretation of factors There is no single solution for handling missing data. Depending on the available data and context, another approach may be needed.

Part II: Factor Analysis with SPSS Slide 21 Item battery "General Health Questionnaire" (GHQ-12) in the Health Survey for England 2003 First Step: Select the Variables Theoretically relevant variables Sufficient number of variables Not too small a sample Descriptive analysis Correlation matrix Scale GHQ-12: All plausible items OK Assumption: 2 factors with 6 variables each Psychological distress Positive health OK Very large sample (n = 8,833) OK See histograms (see Part I). In an ideal situation, normally distributed variables. See below How can dimensions (factors) be discovered? From slide 6: Slide 22 A factor analysis deals with how different items relate to each other, and how they can be grouped into factors. The goal is to group items together as a factor and to replace them with general terms. The general term reflects the underlying content. Each factor represents several items. It is more efficient to represent something through fewer factors than through many single items. Attention: Theoretical and empirical facts are incorporated into a factor analysis. Basic idea of factor analysis Assumption: Some variables tend to be related. Example GHQ-12: ghqstrai ("felt constantly under strain") ghqconfi ("been losing confidence in self") Three possible causes for the ghqstrai ghqconfi correlation: Variable ghqstrai influences variable ghqconfi. Variable ghqconfi influences variable ghqstrai. Both variables are influenced by a factor.

Correlations matrix of the variables AnalyzeCorrelateBivariate C Slide 23 Correlations b ghqsleep ghqstrai ghqover ghqworth ghqunhap ghqconfi ghqhappy ghqface ghqconc ghquse ghqdecis ghqenjoy ghqsleep Pearson correlation coefficient 1.545 **.483 **.387 **.547 **.459 **.343 **.312 **.330 **.228 **.226 **.297 ** Significance (2-sided).000.000.000.000.000.000.000.000.000.000.000 Pearson correlation coefficient.545 ** 1.598 **.418 **.592 **.492 **.376 **.365 **.352 **.217 **.268 **.351 ** ghqstrai Attention: For teaching purposes, Significance (2-sided).000.000.000.000.000.000.000.000.000.000.000 Pearson correlation coefficient.483 **.598 ** 1.517 **.593 **.582 the variables **.397 **.443 are **.369 grouped **.344 together **.345 **.379 ** ghqover Significance (2-sided).000.000.000.000.000.000.000.000.000.000.000 Pearson correlation coefficient.387 **.418 **.517 ** 1.574 **.689 so as **.449 to better **.395 observe **.319 **.362 the structure. This is not normally the case! **.330 **.323 ** ghqworth Significance (2-sided).000.000.000.000.000.000.000.000.000.000.000 ghqunhap Pearson correlation coefficient.547 **.592 **.593 **.574 ** 1.671 **.502 **.428 **.362 **.321 **.308 **.377 ** Significance (2-sided).000.000.000.000.000.000.000.000.000.000.000 ghqconfi Pearson correlation coefficient.459 **.492 **.582 **.689 **.671 ** 1.460 **.427 **.379 **.368 **.355 **.363 ** Significance (2-sided).000.000.000.000.000.000.000.000.000.000.000 ghqhappy Pearson correlation coefficient.343 **.376 **.397 **.449 **.502 **.460 ** 1.487 **.370 **.368 **.380 **.416 ** Significance (2-sided).000.000.000.000.000.000.000.000.000.000.000 ghqface Pearson correlation coefficient.312 **.365 **.443 **.395 **.428 **.427 **.487 ** 1.417 **.389 **.485 **.469 ** Significance (2-sided).000.000.000.000.000.000.000.000.000.000.000 ghqconc Pearson correlation coefficient.330 **.352 **.369 **.319 **.362 **.379 **.370 **.417** 1.389 **.438 **.440 ** Significance (2-sided).000.000.000.000.000.000.000.000.000.000.000 ghquse Pearson correlation coefficient.228 **.217 **.344 **.362 **.321 **.368 **.368 **.389 **.389** 1.488 **.439 ** Significance (2-sided).000.000.000.000.000.000.000.000.000.000.000 ghqdecis Pearson correlation coefficient.226 **.268 **.345 **.330 **.308 **.355 **.380 **.485 **.438**.488 ** 1.367 ** Significance (2-sided).000.000.000.000.000.000.000.000.000.000.000 ghqenjoy Pearson correlation coefficient.297 **.351 **.379 **.323 **.377 **.363 **.416 **.469 **.440**.439 **.367 ** 1 Significance (2-sided).000.000.000.000.000.000.000.000.000.000.000 **. The correlation is significant at the 0.01 level (2-sided). b. Listwise N=8188 There could be two factors (blue, green), where ghqhappy and ghqface cannot be clearly assigned. But: We cannot decide on the number of factors on the basis of a correlation matrix. => Perform a factor analysis! What is important about evaluating the correlation matrix? Slide 24 Significance levels of the correlations The significance level shows whether the correlation coefficient is merely different from zero by chance, or whether there is a high probability that it is truly different from zero. The significance level should be chosen at the beginning of the study (1% or 5%). This depends on the sample size and the goals of the analysis. On rare occasions, especially with large sample sizes, the significance level 0.1% is chosen. Values of the correlation coefficients A factor analysis is problematic when there are many low, yet no high correlation coefficients present. In this case, the data structure is too heterogeneous. Best would be if there are clusters of highly correlated variables that are separated. These clusters are indication of an underlying factor structure.

Second Step: Test of Suitability and Extraction of Factors AnalyzeDimension ReductionFactor F Slide 25 Optional Inverse of correlation matrix: Evaluating suitability of data I Slide 26 The correlation structure is suitable for a factor analysis if its inverse approaches a diagonal matrix. Essentially, it acts as a visual aid. Evaluation: There are no universal rules! A matrix is diagonal if the non-diagonal values are as close as possible to zero. That means that the non-diagonal values should be much smaller than those on the diagonal. Example GHQ-12 The non-diagonal values are significantly smaller than the values on the diagonal. The correlation structure is well-suited for a factor analysis. Diagonal

Bartlett's test: Evaluating suitability of data II Null hypothesis H 0 : The sample is drawn from a population in which all variables are completely uncorrelated. Slide 27 Assumption: The data are normally distributed. Example GHQ-12 In the case of the GHQ-data, Bartlett's test is significant (Sig. =.000) and correspondingly the null hypothesis can be rejected. The variables are not completely uncorrelated. So the factor analysis can be continued. Attention: The statement "The variables are correlated." is false. The alternative hypothesis cannot be postulated. Kaiser-Meyer-Olkin (KMO): Evaluating suitability of data III Kaiser, Meyer and Olkin developed a "Measure of Sampling Adequacy" (MSA), which is the standard test for assessing whether the data are suitable for factor analysis. MSA values refer to single variables. The KMO index is a generalization for the entire dataset. The KMO index shows the extent to which the variables belong together and so helps to determine whether or not a factor analysis is appropriate. It tests whether the partial correlations between the variables are small. If these are small, then the KMO is high. Rule of thumb: The KMO should be.60 or higher in order to continue with the factor analysis. Kaiser (1970) suggests a lower limit of.50, though a value of.80 or higher would be desirable. Slide 28 Example GHQ-12 KMO value.00 to.49 unacceptable.50 bis.59 miserable.60 bis.69 mediocre.70 bis.79 middling.80 bis.89 meritorious.90 bis 1.00 marvellous

Example GHQ-12: Comparing the extraction results Slide 29 Factor loading Factor loadings The factor loading of a variable is the correlation between the variable and the factor. Typical statement: "The variable ghqconc loads with.619 on factor 1." Values between -1 and +1 are theoretically possible. The magnitude of the factor loading shows how closely a variable correlates with a factor: Magnitudes close to 0 indicate that a relationship barely exists. The higher the magnitude, the closer the relationship. Graphical interpretation of the factor loadings Each variable can be described as a vector in a coordinate system. The vector is formed by the factor loadings of the variable. The factor loadings can be interpreted as coordinates. Slide 30 1.0 Factor 2 0.8 0.6 0.4 0.2.118 0.0-1.0-0.8-0.6-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0-0.2-0.4 ghquse ghqdecis ghqenjoy ghqconc ghqface ghqhappy Factor 1.679 ghqworth ghqconfi ghqover ghqunhap ghqsleep ghqstrai -0.6-0.8-1.0

Communality The variables cannot usually be completely explained by the factors. Slide 31 Communality is the total amount of a variable's variance that can be explained by all factors. Communality indicates the extent to which a variable is explained through the factors. Example GHQ-12 Example variable ghqconq: Communality after extraction.481 48.1% of the variance of ghqconq is explained through factors 1 and 2. Association with Component Matrix 0.481 =.619 2 +.314 2 48.1% = 38.3% + 9.8% Third Step: Determining the Number of Factors Slide 32 There is no unambiguous method for determining the number of factors. Common sense: Limit the number of factors to those which are understood. The number of factors is usually significantly less than the number of variables. Kaiser Criterion The Kaiser criterion removes all components whose eigenvalue is under 1.0. It is the default in SPSS and most other statistical programs. The Kaiser criterion, however, is not recommended as the only basis for a decision. Scree Plot The scree plot shows the components along the x-axis and the corresponding eigenvalues along the y-axis. All components past the elbow are omitted. Explained variance as the criterion Some researchers apply the rule that there be a sufficient number of factors to describe 90% (sometimes 80%) of the variance.

Slide 33 Kaiser criterion (Eigenvalues > 1.0) An eigenvalue shows how much of the total variance is explained by the factor. The Kaiser criterion requires that all components with an eigenvalue < 1.0 be rejected. The Kaiser criterion is not recommended as the only basis for a decision, since it commonly leads to choosing too many factors. Example GHQ-12 The eigenvalue corresponds to the proportion of the total variance that is explained by the factor. The variables are z-transformed for this purpose (standard deviation 1 and mean 0). Thus the total variance of the GHQ-12 to be explained (12 variables) = 12. The first factor explains 5.588 of this amount, and so 5.588/12 (46.568%) of the variance. The second factor explains 1.330/12 (11.082%) of the variance. 2 factors at most Kaiser criterion (Eigenvalue > 1.0) Scree Plot: Example GHQ-12 Slide 34 The elbow occurs at 3 2 factors Typical statement: elbow flat slope I have decided on two factors, because this is theoretically plausible and consistent with the Kaiser criterion (eigenvalue > 1) as well as the scree plot. Scree plot: Example with random data No elbow Elbow criterion: If the factors arise randomly, then the slope is flat. Only factors above the elbow are counted. http://en.wikipedia.org/wiki/scree Scree

Fourth Step: Interpretation of the Factors Rotation Rotation leads to a better readability of the results. Varimax rotation preserves the independence of the factors. The factors are rotated, so that the variance of the squared loadings per factor is maximized. Mid-level loadings tend to become "smaller" or "larger", so that the factor structure is easier to interpret. Slide 35 Symbolic representation: Before rotation: After rotation: Factor 1 Factor 1 Factor 2 Factor 2 Example GHQ-12: Rotation and loading plots in SPSS Loading plots essentially constitute a visualization. Slide 36 Before rotation: After rotation:

Evaluating the loadings and assigning variables to factors Slide 37 In principle, each variable is assigned to the factor on which it loads highest. How high should a factor loading minimally be in order to be interpreted? There are different possible rules of thumb: Factor loadings lower than.20 should not be considered. If an item does not load any higher on any of the factors: Discard the item and redo the analysis. Factor loadings of ±.30 to ±.40 are minimally acceptable but higher values are desirable (particularly if the sample is small and only few variables are considered) Irrespective of sample size: A factor can be interpreted ifc at least 4 variables show loadings of.60. at least 10 variables have a loading of.40. If n < 300, factors with low loadings only should not be interpreted. Example GHQ-12 The factors show 6 and 5 variables with loadings.6, respectively. Cross-loadings Slide 38 A cross-loading occurs if a variable has high loadings on more than one factor. (two or more factor loadings >.3 or >.4) The item is related with more than one factor. It correlates with other items that load on the affected factors. If the goal is to find sharply-defined factors, and from them, for example, form sum scores, the difference between loadings is considered. Large difference (>.2): The variable can be assigned to the factor with the highest loading. Small difference (<.2): The variable cannot be assigned to a factor. Exclude and perform the analysis again. (Unless theoretical considerations contradict this.) If the goal is to demonstrate commonality of concepts, cross-loadings are of theoretical interest and are retained. They must be theoretically plausible.

Content interpretation of the factors I Slide 39 The items are sorted within the factor according to the decreasing magnitude of the loading. SPSS can facilitate doing so. Not sorted: Sorted by size: Slide 40 Content interpretation of the factors II The sign of the loading (or the loading itself) is noted. What is the "theme" of the factors? What do the factors stand for? Inspect item text. The highest-loading variables are especially helpful (so-called "marker variables"). The context of the study can also provide information. Example GHQ-12 ghqunhap Been feeling unhappy and depressed.810.261 ghqstrai Felt constantly under strain.769.164 ghqconfi Been losing confidence in self.747.328 ghqover Felt couldn t overcome difficulties.730.304 ghqsleep Lost sleep over worry.728.124 ghqworth Been thinking of self as worthless.671.320 ghqdecis Felt capable of making decisions.128.764 ghquse Felt playing useful part in things.133.743 ghqface Been able to face problems.331.665 ghqenjoy Able to enjoy day-to-day activities.259.663 ghqconc Able to concentrate.267.640 ghqhappy Been feeling reasonably happy.440.530 Factor 1 Factor 2 Possible general term: "psychological distress" "positive health"

Fifth Step (Optional): Calculate Sum Scales or Factor Scores To use factors in further analyses, we need to generate one variable per factor to represent it. Usually, sum scales are calculated on the basis of the factor solution. Alternatively, factor scores can be calculated. Slide 41 Calculating sum scales in SPSS TransformCompute variablesc Test each factor for reliability beforehand! (Cronbach's alpha) Calculating factor scores in SPSS It is implemented as part of SPSS factor analysis (Option "Scores"). The "regression method" is most often used. This results in one variable per factor, e.g. FAC1_1, FAC1_2. References Hankins, M. (2008) The factor structure of the twelve item General Health Questionnaire (GHQ-12): the result of negative phrasing? Clinical Practice and Epidemiology in Mental Health, 4:10, www.cpementalhealth.com/content/4/1/10 (Date of access: February 2014) Slide 42 Jackson, C. (2007) The General Health Questionnaire. Occupational Medicine, 57:79, http://occmed.oxfordjournals.org/content/57/1/79.full (Date of access: February 2014) Kaiser, H. (1970). A second generation little jiffy. Psychometrika, 35(4), 401-415. www.ats.ucla.edu/stat/spss/faq/alpha.html (Date of access: February 2014)

Appendix Slide 43 Types of Factorizations: Principal Components Analysis vs. Principal Axis Factoring Principal Component Analysis (PCA, default in SPSS) Goal: Reproduce the data structure No causal relationship between factors and variables Factors are "general terms" and are often called components. Process: The first factor is chosen so that it describes the greatest possible proportion of variance in the variables. Each additional factor describes a maximum amount of the remaining variance. Factors are extracted until the total variance among the variables is explained. If variables are added, the factor loadings change. Principal Axis Factoring (PAF) Goal: Determine the cause of the correlation structure Causal interpretation: Factors cause the correlations among variables. Process: The first factor is chosen so that it explains the greatest possible portion of the common variance of the variables. Each additional factor explains a maximum portion of the remaining common variance of the variables. Factors are extracted until the factors can explain all the common variance in a set of variables. In principle it is possible to add variables without affecting the factor loadings. Notes: Slide 44