Formatting Factor Analysis Results as a Set of Tables in Research Reports Factor analysis provides you with a lot of output, and you may not have prior experience of reporting the results of a factor analysis in a research report. Published articles are often only allowed restricted space for tables, so they may not always provide useful examples. Hence, this document is to provide you with an example of how to report your factor analysis. Note that this document is not supposed to be exhaustive, and nor is it to train you in conducting a factor analysis or identifying the appropriate parts of the output. The lecture material and workbook activities will provide you with that information. This document simply provides you with a factor analysis table format that will be useful for the main assignment, and potentially for any further study of future work that involves doing factor analysis. 1. The Factor Analysis On the following pages are three tables that summarise a factor analysis on the 7 Up 7 Down (7U7D; Youngstrom et al., 2013), a questionnaire measure of trait bipolar disorder vulnerability. Your factor analysis tables should be formatted similarly to these. Important points to note: o It usually makes more sense to present tables shortly after introducing them in-text, and to describe what is summarised in them. This allows the reader to quickly connect the results with your description. o DO NOT present statistics in a table and then also present the same statistics in-text. o Fitting tables into a document and making them look nice can be difficult. The APA guide does not prescribe a given font size or line-spacing for tables, as a one-size-fits-all approach would not work. The following tables are formatted them in Size 11 font, and Multiple: 1.2 line-spacing. The notes below the first two tables are in Size 12 font. o Writing out the items takes up a lot of space, but it can be confusing to talk about the items if they are not present somewhere in the document. In Table 1 the items are in Size 10 font so that they were a little more compact which helps the Table to fit on one page. o Another solution would be to present a table of items in the Materials section of your Method, or to have a list of items in an Appendix at the end. This can be especially useful if you are presenting a factor solution where you have deleted items (no items were deleted in this example).  If you have a table in the Method or an Appendix, in your Results you could say something like, “Item 15 (see Appendix A) was removed because….” By referring to the Appendix you don’t have to write out the full item wording every time you mention that item. o The factor loadings are presented to two decimal places (about the only statistic that you should present to three decimal places in any report is a p-value). When you are trying out different factor analyses, you may use the “supress” function in SPSS to blank out loadings below your factor loading cut-off. However, when reporting, it’s best to show all of the loadings on each factor. Some papers, as in Table 1, present the highest loadings for each item in bold text to make it clear which factor an item is loading on.This is not compulsory though. o Something that can also be helpful is to clump items loading on a given factor together in a table; for example, all of the depression-proneness items first, and then all of the mania-proneness items. This was not done in Table 1 as the items are from a published scale with an established item order. Example Factor Analysis Tables: Table 1 7U7D Pattern Matrix obtained via Maximum Likelihood Extraction with Promax Rotation Item Factor 1: Depression-proneness Factor 2: Mania-proneness 1. Have you had periods of extreme happiness and intense energy lasting several days or more when you also felt much more anxious or tense (jittery, nervous. uptight) than usual (other than related to the menstrual cycle)? .14 .65 2. Have there been times of several days or more when you were so sad that it was quite painful or you felt that you couldn't stand it? (GBI 23) .69 .15 3. Have there been times lasting several days or more when you felt you must have lots of excitement, and you actually did a lot of new or different things? -.11 .78 4. Have you had periods of extreme happiness and intense energy (clearly more than your usual self) when, for several days or more, it took you over an hour to get to sleep at night? -.04 .76 5. Have there been long periods in your life when you felt sad, depressed, or irritable most of the time? .83 .05 6. Have you had periods of extreme happiness and high energy lasting several days or more when what you saw, heard, smelled, tasted, or touched seemed vivid or intense? -.03 .77 7. Have there been periods of several days or more when your thinking was so clear and quick that it was much better than most other people's? -.02 .62 8. Have there been times of a couple days or more when you felt that you were a very important person or that your abilities or talents were better than most other people's? .03 .57 9. Have them been times when you have hated yourself or felt that you were stupid, ugly, unlovable, or useless? .85 -.11 10. Have there been times of several days or more when you really got down on yourself and felt worthless? .89 -.04 11. Have you had periods when it seemed that the future was hopeless and things could not improve? .86 -.03 12. Have there been periods lasting several days or more when you were so down in the dumps that you thought you might never snap out of it? .87 .01 13. Have you had times when your thoughts and ideas came so fast that you couldn't get them all out, or they came so quickly that others complained that they couldn't keep up with your ideas? .19 .57 14. Have there been times when you have felt that you would be better off dead? .75 .04 Eigenvalues 6.92 2.09 Extraction SSL 6.44 1.74 Rotation SSL 5.93 4.90 Note. SSL = sum of squared loadings (initial SSL are equivalent to the eigenvalues). o Table 2 notes the Cronbach’s alpha values for the subscales in the table note. This is because the Cronbach’s alpha if item deleted statistics would be fairly meaningless without this reference point. o Note that Cronbach’s alphas for each subscale are also mentioned in Table 3. It’s generally poor form to present the same statistics multiple times. It would be best to only present this information in either table, or not in any table. For example, you could describe the relevant alphas for the new scale in text before introducing Table 2.  The Cronbach’s alpha values and corrected item-total correlations do not come from the factor analysis output, but from running separate reliability analyses on the relevant items.  Communalities come from the factor analysis output  M(SD) item response values come from running Explore or Descriptives o Tables 1 and 2 could possibly be merged if the page orientation was changed to landscape format. Either way is fine. You’ll need to learn to apply section breaks in order to use landscape orientation, and this is a useful skill to have for managing page numbering in complex documents more generally. Table 2 Item-Level Properties of the 7U7D obtained via Exploratory Factor Analysis Item Communalities (extraction) Item response M(SD) Corrected item-total correlation Cronbach’s α if item deleted Item 1 (M) .54 0.61 (.79) .65 .84 Item 2 (D) .62 0.79 (.90) .75 .93 Item 3 (M) .53 0.76 (.81) .65 .84 Item 4 (M) .55 0.73 (.87) .66 .84 Item 5 (D) .73 0.99 (.99) .82 .92 Item 6 (M) .57 0.49 (.77) .68 .84 Item 7 (M) .37 0.80 (.81) .59 .85 Item 8 (M) .34 0.65 (.80) .56 .85 Item 9 (D) .63 1.04 (.94) .75 .93 Item 10 (D) .76 0.93 (.92) .83 .92 Item 11 (D) .71 0.90 (.90) .81 .93 Item 12 (D) .76 0.80 (.91) .84 .92 Item 13 (M) .48 0.59 (.82) .64 .84 Item 14 (D) .60 0.68 (.89) .75 .93 Note. M = mania-proneness item, D = depression-proneness item. Mania-proneness Cronbach’s α = .86, depression-proneness Cronbach’s α = .94. 2.Descriptives Once you have finished reporting your factor analysis, you need to move from an item-level of data to a scale-level of data. This means that your next step is to tally up whole-scale and subscale scores for your new scale. You can do this by using SPSS (Transform > Compute new variable) to add up the items that have loaded significantly on each factor. For example, to calculate participants’ mania-proneness scores you would compute a variable that summed Items 1, 3, 4, 6, 7, 8, and 13 of the 7U7D. Another option is to sum and average the scores (i.e., add up those items and divide by 7). Once this is done you should report descriptive statistics for your new scale, just like you would in any research report. Since descriptive statistics for your validity indicators would be useful for the reader, you should include them here as well. Example Descriptives Table: Table 3 Descriptive Statistics for the 7U7D and Validity Indicators M (SD) Actual range Potential range Skew Kurtosis Cronbach’s α 7U7D Mania-proneness 3.82 (3.68) 0-18 0-18 1.03 .62 .86 Depression- proneness 6.13 (5.50) 0-21 0-21 0.92 .12 .94 Validity indicators BAS-D 10.64 (2.49) 4-16 4-16 0.08 -.16 .81 BAS-FS 11.41 (2.40) 4-16 4-16 -0.22 -.01 .74 BAS-RR 15.98 (2.61) 5-20 5-20 -0.56 .52 .78 Trait BIS 12.11 (2.45) 4-16 4-16 -0.40 -.01 .75 N = 760 Note. Standard deviations are presented in parentheses following the relevant mean. Important points to note: o You might be wondering why no data for a “7U7D total” score here, as in many cases subscales would be summed to provide a total scale score. In this case, the assumption is that depression-proneness and mania-proneness are theoretically distinct. If they were added them together and we then ran correlations using 7U7D total, we could be obscuring a lot of important data about how depression-proneness and mania-proneness relate differently to other constructs. When designing a new scale, it is up to you to think about whether your subscales can and should be meaningfully combined or not. o The actual range is the minimum to maximum values that you observed in your data set. The potential range is the minimum to maximum values that it is possible to obtain on the measure. These values can be found by multiplying the number of items by the lowest scale anchor (minimum possible response) and by the highest scale anchor (maximum possible response). o For example, if you have 10 items on a 1 (Strongly Disagree) to 10 (Strongly Agree) scale, the potential range is 10-100.  You should always be aware of the potential range, as without knowing this you cannot spot out-of-range values (errors) in your data set. o In Table 3 the items have been summedto obtain subscale scores, which is reflected by the range values. An alternative to this is to average your scale items. This can help to make the data more interpretable at face value. o For example, 17 items measured on a 7 point scale (1 = Strongly Disagree, 7 = Strongly agree) has a potential range of 17-119, and might have an actual range of 25-108. The average potential range is 1-7, and the average actual range is 3.57-6.35. This can make it easier to think about what the results mean based on the Strongly Disagree/Strongly Agree response format. o Skew and kurtosis can be found through the Descriptives> Explore function. o Cronbach’s alpha needs to calculated separately (see Week 3 workbook activity for a guide on this).   3. Table of Subscale and Validity Correlations Now that you have presented descriptives, it is time to present your validity correlations. Since it would also be useful to the reader to know how your new scale or subscales are correlated, you might as well present this information at the same time. Here are two examples of how you could tabulate a correlation matrix. The first example is specific to our scale validation purpose, as often for a scale validation we don’t really care about how the validity indicators are correlated with each other, so we can chop this information off to tell a more parsimonious story. The second example is the classic “bottom triangle along the main diagonal” that we all know and love. You should only use one of these examples, not both. Important points to note: o The results in these tables are based on bivariate correlations between scale scores that have been created in SPSS AFTER completing the factor analysis. The matrix of factor correlations that are given in the factor analysis output is a bit different to that obtained by running bivariate correlations. This is because it runs correlations using item data modified by the factor loadings. Sometimes strange things can occur due to this, such as correlations that you would expect to be positive appearing negative. Often abnormalities such as this disappear when running bivariate correlations, so it is generally better to focus on bivariate correlations. o The Tables are a bit different to most validations as theyonly look at two scales, rather than having a bunch of different validity indicators. This is because the 7U7D was only compared to the BIS/BAS scales in this dataset. However, the same formatting logic would apply in a more conventional scale validation study. o We’re accomplishing two purposes with this table: we’re (i) telling the reader how our subscales are correlated; and (ii) testing the validity of our scale and/or subscales. If you have a lot of subscales or validity indicators and feel that this table format is becoming unwieldy, you can of course separate them into two separate tables. o If we had a meaningful “7U7D total” variable, then usually we would probably use that as a focus when testing our validation hypotheses, with the subscale data simply providing an extra level of richness. However, as noted previously, depression-proneness and mania-proneness are being considered as fairly different constructs that are not strongly correlated enough for a “7U7D total” variable to be meaningful. o Note that whilst a total variable might be useful for testing against validity indicators, which is why it might put it in a table of validity correlations, comparing subscales to a total doesn’t really tell us much.It would normally be statistically redundant to do this, because the total is partially composed of the subscale that you are comparing to it. Examples of Two Different Validity Table Formats: Table 4a [FOCUSED EXAMPLE] Intercorrelation between the 7U7D and BIS/BAS Scales 1. 2. 1. Mania-proneness – 2. Depression-proneness .54*** – 3. BAS-D .11** -.12** 4. BAS-FS .13*** -.10* 5. BAS-RR -.11** -.19*** 6. Trait BIS -.09 .21*** N = 760 Note: * p< .05, ** p< .01, *** p< .001 Table 4b [STANDARD EXAMPLE] Intercorrelation between the 7U7D and BIS/BAS Scales 1. 2. 3. 4. 5. 6. 1. Mania-proneness – 2. Depression-proneness .54*** – 3. BAS-D .11** -.12** – 4. BAS-FS .13*** -.10* .47*** – 5. BAS-RR -.11** -.19*** .51*** .52*** – 6. Trait BIS -.09 .21*** .07 .15*** .39*** – N = 760 Note: * p< .05, ** p< .01, *** p< .001