Technical Whitepaper
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1 Technical Whitepaper July, 2001 Prorating Scale Scores Consequential analysis using scales from: BDI (Beck Depression Inventory) NAS (Novaco Anger Scales) STAXI (State-Trait Anxiety Inventory) PIP (Psychotic Inpatient Profile) WARS (Ward Anger Rating Scale) Paul Barrett 24 th May, 2017 This research was conducted in 2001, while the first author was a Research Associate, and second author the Chief Research Scientist at the High Security Forensic Psychiatric State Hospital, Carstairs, Scotland, UK. The work provides some detailed data and graphics on the consequences of prorating scale scores. It may be helpful for those looking for some empirically justified insights in this area, especially when small sample sizes contraindicate multiple imputation methods.
2 The State Hospital Anger Project This analysis, implemented by Charles Marley, investigated the accuracy of using prorated scale scores with a variety of scales from the State Hospital Anger Management project. The scales were chosen as representative of the item length and internal consistency of all the scales used in the project. The aim is to permit a rational examination of the problem, and to decide upon a common fixed lower bound for all the tests, based upon the analysis results below. The only alternative to this approach (which still yields some patients with unscored scales), is to use multiple imputation of the missing item data, using either the AMELIA or NORM programs (using the EM routine) to impute the most likely missing value for every occurrence. This is something we are looking at for the future. Selecting Bound Values for Pro-Rating Missing Items The scales are chosen from five of the more commonly used anger assessments, and all the scales vary in the amount of items that make up the scale. The Scales used and the items that make up the scales are listed below: Assessment Scale No. items Items Alpha Page BDI Total Score NAS Behaviour ; 26-29; 41-44; * 5 NAS Regulation 12 5,10,15,20,25,30,35,40,45,5 0.85* 8 0,55,60 STAXI State Anger * 11 STAXI Anger Expression * 14 PIP Hostile Belligerence 8 11,28,31,36,51,54,67, PIP Disorientation WARS Anger Attributes * 23 WARS Self Aggression ? 26 Data The data was obtained from the State Hospital Anger Admission Project and Anger Intervention Group Files. Each analysis is produced with approximately 60 male patients per assessment. The *alphas are obtained from State Hospital Anger Admission research report. Procedure Followed The number of items to be missing from each scale was set at 5% intervals from 5% to 50% - this allocated different amounts of items to be missing, for each of the scales, based on the number of items in the scale. The scores were obtained for the scale, with the missing items, and then the pro-rated score was calculated using the following equation: T p T N R R * N T where T p = Prorated Total Scale Score T R = Total Raw Score based upon those items which have been responded to N R = Number of items which have been responded to N T = The total number of items in the scale The prorated scores were then subtracted from the total score, (data obtained from the admissions data file except for the BDI data, which was obtained from the anger group data file), to obtain the differences. The prorated scores were then correlated with the each of the prorated scores to produce correlation matrices, and descriptive statistics were calculated for the total score, the prorated score and the differences between both of these scores. Box whisker plots of the median of the differences between the pro-rated total and the true score total, the inter-quartile points of these differences and the minimum and maximum differences are displayed, showing the variability of the scores at different levels of prorating. Note: the item exclusions were semi-random but represent only one out of many possible item selections. Charles Marley and Paul Barrett: The State Hospital, Carstairs 2
3 BDI Total Score Missing Items Percentage (%) value of 21 items No. of items excluded Items excluded , ,14, ,5,9, ,5,6,13, ,4,5,7,9, ,8,11,15,18,19, ,2,8,12,15,18,20, ,7,8,11,12,15,17,19, ,2,5,9,12,15,16,19,20,21 BDI Total Score - After Item Exclusion and Pro-Rating Charles Marley and Paul Barrett: The State Hospital, Carstairs 3
4 BDI Total Score - Descriptive Statistics after Item Exclusion BDI Total Score - Descriptive Statistics after Pro-Rating BDI Total Score / Pro-Rated Value Descriptive Statistics of Differences BDI Mean Total / BDI Pro-Rated Mean Correlation Charles Marley and Paul Barrett: The State Hospital, Carstairs 4
5 BDI Total - Mean Differences between Pro-Rated and Total Scores BDI Total Score - Pro-Rated Scale Total / Scale Total Differences Variability Charles Marley and Paul Barrett: The State Hospital, Carstairs 5
6 NAS Behaviour Total Score Missing Items Percentage (%) value of 15 items No. of items excluded Items excluded , ,12, ,26,29, ,27,41,56, ,14,29,43,56, ,29,41,44,56,57, ,14,28,29,41,44,56,57 NAS Behaviour Score - After Item Exclusion and Pro-Rating Charles Marley and Paul Barrett: The State Hospital, Carstairs 6
7 NAS Behaviour (Mean) Score after Item Exclusion NAS Behaviour Total - Descriptive Statistics after Pro-Rating NAS Behaviour (Mean) Total / Pro-Rated Total Descriptive Statistics NAS Behaviour Total / Pro-Rated Total Correlation Charles Marley and Paul Barrett: The State Hospital, Carstairs 7
8 NAS Behaviour (Mean) Total / Pro-Rated Score Differences NAS Behaviour Scale Pro-Rated Scale Totals / Scale Totals Differences - Variability Charles Marley and Paul Barrett: The State Hospital, Carstairs 8
9 NAS Regulation Total Score Missing Items Percentage (%) value of 12 items No. of items excluded Items excluded , ,30, ,40,50, ,30,40,50, ,15,35,40,45,55 NAS Regulation Score - After Item Exclusion and Pro-Rating Charles Marley and Paul Barrett: The State Hospital, Carstairs 9
10 NAS Regulation Scale after Item Exclusion Descriptive Statistics NAS Regulation Total after Pro-Rating Descriptive Statistics NAS Regulation Total /Pro-Rated Total (Mean) Differences Descriptive Statistics NAS Regulation Total / Pro-Rated Total Correlation Charles Marley and Paul Barrett: The State Hospital, Carstairs 10
11 NAS Regulation Total Score / Pro-Rated Score - Mean Differences NAS Regulation Scale Pro-Rated Totals / Total Score Differences - Variability Charles Marley and Paul Barrett: The State Hospital, Carstairs 11
12 STAXI State Anger Total Score Missing Items Percentage (%) value of 10 items No. of items excluded Items excluded , ,5, ,4,7, ,2,5,7,9 STAXI State Anger Total After Item Exclusion and Pro-Rating Charles Marley and Paul Barrett: The State Hospital, Carstairs 12
13 STAXI State Anger Total after Item Exclusion Descriptive Statistics STAXI State Anger Total After Pro-Rating Descriptive Statistics STAXI State Anger Total / Pro-Rating Total Differences Descriptive Statistics STAXI State Anger Total / Pro-Rating Totals Correlation Charles Marley and Paul Barrett: The State Hospital, Carstairs 13
14 STAXI State Anger Total / Pro-Rating Total Differences STAXI State Anger Scale Pro-Rate Totals / Total Score Differences - Variability Charles Marley and Paul Barrett: The State Hospital, Carstairs 14
15 STAXI Anger Expression Total Score Missing Items Percentage (%) value of 24 items No. of items excluded Items excluded , ,35,38, ,25,37,38, ,25,27,31,40, ,25,26,35,38,40, ,28,30,33,36,40,41, ,23,24,25,31,32,35,41,42, ,23,25,28,31,33,37,38,41,42,43, ,23,25,26,29,32,34,36,38,40,42,43 STAXI Anger Expression Total After Item Exclusion and Pro-Rating Charles Marley and Paul Barrett: The State Hospital, Carstairs 15
16 STAXI Anger Expression Total After Item Exclusion Descriptive Statistics STAXI Anger Expression Total After Pro-Rating Descriptive Statistics STAXI Anger Expression Total / Pro-Rated Total Differences Descriptive Statistics STAXI Anger Expression Total / Pro-Rated Total Correlation Charles Marley and Paul Barrett: The State Hospital, Carstairs 16
17 STAXI Anger Expression Total / Pro-Rating Total Differences STAXI Anger Expression Scale Pro-Rate Total / Total Score Differences - Variability Charles Marley and Paul Barrett: The State Hospital, Carstairs 17
18 PIP Hostile Belligerence Total Score Missing Items Percentage (%) value of 8 items No. of items excluded Items excluded , ,67, ,28,36,51 PIP Hostile Belligerence Total After Item Exclusion and Pro-Rating Charles Marley and Paul Barrett: The State Hospital, Carstairs 18
19 PIP Hostile Belligerence Total after Item Exclusion Descriptive Statistics PIP Hostile Belligerence Total after Pro-Rating Descriptive Statistics PIP Hostile Belligerence Total / Pro-rating Total Differences Descriptive Statistics PIP Hostile Belligerence Total / Pro-Rating Total Correlation Charles Marley and Paul Barrett: The State Hospital, Carstairs 19
20 PIP Hostile Belligerence Total / Pro-Rating Total Differences PIP Hostile Belligerence Scale Pro-Rate Total / Total Score Differences - Variability Charles Marley and Paul Barrett: The State Hospital, Carstairs 20
21 PIP Disorientation Total Score Missing Items Percentage (%) value of 5 items No. of items excluded Items excluded , ,94,95 PIP Disorientation Total After Item Exclusion and Pro-Rating Charles Marley and Paul Barrett: The State Hospital, Carstairs 21
22 PIP Disorientation Total after Item Exclusion Descriptive Statistics PIP Disorientation Total After Pro-rating Descriptive Statistics PIP Disorientation Total / Pro-Rating Total Differences Descriptive Statistics PIP Disorientation Total / Pro-Rating Total Correlation Charles Marley and Paul Barrett: The State Hospital, Carstairs 22
23 PIP Disorientation Total / Pro-Rating Total Differences PIP Disorientation Scale Pro-Rate Total / Total Score Differences - Variability Charles Marley and Paul Barrett: The State Hospital, Carstairs 23
24 WARS Anger Attributes Total Score Missing Items Percentage (%) value of 7 items No. of items excluded Items excluded , ,21, ,20,21,23 WARS Anger Attributes Total After Item Exclusion and Pro-Rating Charles Marley and Paul Barrett: The State Hospital, Carstairs 24
25 WARS Anger Attributes (Mean) Total after Item Exclusion Descriptive Statistics WARS Anger Attributes Total after Pro-Rating Descriptive Statistics WARS Anger Attributes Total / Pro-Rated Total Differences Descriptive Statistics WARS Anger Attributes Total / Pro-Rated Total Correlation Charles Marley and Paul Barrett: The State Hospital, Carstairs 25
26 WARS Anger Attributes Total / Pro-Rated Total Differences WARS Anger Attributes Scale Pro-Rate Total / Total Score Differences - Variability Charles Marley and Paul Barrett: The State Hospital, Carstairs 26
27 WARS Self Aggression Total Score Missing Items Percentage (%) value of 4 items No. of items excluded Items excluded ,15 WARS Self-Aggression Total After Item Exclusion and Pro-Rating Charles Marley and Paul Barrett: The State Hospital, Carstairs 27
28 WARS Self Aggression Total after Item Exclusion Descriptive Statistics WARS Self Aggression Total after Pro-Rating Descriptive Statistics WARS Self Aggression Total / Pro-Rating Total Differences Descriptive Statistics WARS Self Aggression Total / Pro-Rating Total Correlation Charles Marley and Paul Barrett: The State Hospital, Carstairs 28
29 WARS Self Aggression Total / Pro-Rating Total Differences WARS Self Aggression Scale Pro-Rate Total / Total Score Differences - Variability Charles Marley and Paul Barrett: The State Hospital, Carstairs 29
30 Discussion Points When pro-rating missing items it would seem possible to correlate up to 50% of the items based on the high correlation co-efficient. This is partly true. When applying statistical techniques that analyse the covariance between scales, or the covariance of one of the scales with a separate measure, it would be appear acceptable to pro-rate substantive amounts of missing data. The correlation matrix for each of the scales shows high levels of correlation between the total score and up to 50% of the pro-rated scores. This is due to the total score and the pro-rated score being monotonically related if the total score is high the pro-rated score will also be proportionately high and as the true score gets higher the pro-rated score will also get higher (this is shown by the linear fit of the scores after pro-rating). When looking at the absolute value of the scores and the reproducibility of these scores through prorating, other factors have to be taken into account, as product-moment (pearson) correlation will be insensitive to these actual values. The box-whisker plots display the variability of the absolute value of the score after pro-rating. The inter-quartile range shows the 50% of the data can be situated 2-3 points above or below the median but the minimum and maximum values of the data can be up to ten points above or below the median on certain levels of pro-rating. This could make a difference in the classification of an individual on the particular trait being measured. This is clearly demonstrated when pro-rating 35% of the items in the Beck Depression Inventory (BDI). When 35% of the items are prorated, a difference of 6.5 points above or 7.5 points below the true score, which is obtained with the same data but with no missing items, is reached this could make the difference to a individual being classified as depressed or not. The above two paragraphs make it clear that two issues need to be considered here: prorating scores for use in covariance analysis only, in which by standardising the data prior to computing agreement, the actual size of scores is lost. If say we wish to compute a factor analysis using correlations, or perhaps a standardised regression analysis, then prorating the scores to a larger degree (say up to 50%) may be considered acceptable. However, to prorate such large missing amounts of data for a summated scale score will only be reasonable if the alpha (internal consistency) coefficient is larger than at least 0.80 (which represents a correlation of 0.89 between the observed and true scores). The second issue is the one that we have to face here. We are prorating scores primarily to preserve as many valid cases with test scores as possible. Yet, the actual value of the prorated score will be used as a meaningful score in its own right, and will be interpreted as though the individual had completed all the items within a scale. This means that prorating error can have significant effects on the classification and change-score status for an individual. Of course, as mentioned in the first paragraph on the first page, we might have used multiple mputation for every missing value. From Joe Schafer s text Multiple imputation is a simulation-based approach to the statistical analysis of incomplete data. In multiple imputation, each missing datum is replaced by m>1 simulated values. The resulting m versions of the complete data can then be analyzed by standard complete-data methods, and the results combined to produce inferential statements (e.g. interval estimates or p-values) that incorporate missing-data uncertainty. However, this methodology requires some research attention as it is pretty sophisticated and sometimes computationally intensive. See/download: Longitudinal and multi-group modelling with missing data. Werner Wothke, Smallwaters Corp: and the AMELIA program site at: and the NORM and EMCOV programs (and Joe Schafer s explanations) at: So, the final question to be addressed and answered, is what missing data bound should we use to prorate scale scores for the Anger Management project? The recommendation is a 15% bound across all tests, given the evidence in this document. We are trying to preserve the maximum number of patients data whilst taking care to constrain the error to a tolerable and sensible amount. In fact, this is somewhat generous but, given further analysis, you can see that the error induced is still acceptable for all practical purposes. Charles Marley and Paul Barrett: The State Hospital, Carstairs 30
31 Some justification of the chosen 15% bound. Let s take a few scales BDI Total Score / Pro-Rated Value Descriptive Statistics of Differences A 15% missing bound (3 items) incurs at least one error of and on the relevant true scores. However, the mean difference is just From the associated box-whisker plot of errors, we can see that 50% of the estimation errors lie within a ±1.0 bound. To check this, we computed a frequency distribution of all estimate errors: Category Missing Frequency table: BTD15P: 15% prorated MIN Count Cumulative Percent Cumulative Count Percent Here we see that only two cases have estimates in error to , with just one case at Further, this maximum error of is 10% of the score range (40). Given the ordinal nature and interpretation of the scores, it is felt that such a discrepancy is not likely to be critical. However, this depends upon the actual score originally achieved. The two 4 cases originally scored 32 and 39, with the overestimate occurring with a an original score of 33. If exactitude is required, then no prorating should take place however, it is felt that the degree of error incurred with a 15% prorating is probably acceptable from a clinical perspective, however, this is a ultimately a matter for clinicians to consider. Charles Marley and Paul Barrett: The State Hospital, Carstairs 31
32 STAXI State Anger Total / Pro-Rating Total Differences Descriptive Statistics A 15% missing bound (2 items) incurs at least one error of 2.25 and +2.0 on the relevant true scores. However, the mean difference is just From the associated box-whisker plot of errors, we can see that 50% of the estimation errors lie within a ±0.0 bound. To check this, we computed a frequency distribution of all estimate errors: Category Missing Frequency table: STD15P: 15-20% Prorated Count Cumulative Percent Cumulative Count Percent Once again, the number of cases with the large marginal errors are low (only 2 cases), which again supports the 15% bound. It is of interest to note that as we move to a 25-30% prorating bound (3 items), the error jumps to This is considered unacceptable, or at least undesirable. Category Missing Frequency table: STD25P: 25-30% Prorated Count Cumulative Percent Cumulative Count Percent However, the frequency table for this 3-item (25-30%) prorating is not too bad but the error is large, and for simplicity it is easier to stick with a 15% bound throughout also, these results are dependent upon just one set of item exclusions so cannot be considered definitive. The conservative approach is to limit prorating to just 2 items. WARS Self Aggression Total / Pro-Rating Total Differences Descriptive Statistics Charles Marley and Paul Barrett: The State Hospital, Carstairs 32
33 The WARS scale has just 4 items, with a total score of 4, and no published alpha reliability coefficient. If we exclude just one item, we incur an error of up to 1.0 through to If we exclude two items. Our prorating error becomes 1.00 through to Looking at the distribution of estimation errors, we have: 1 item exclusion prorated Category Missing Frequency table: WSD15P: 15-35% Prorate Count Cumulative Percent Cumulative Count Percent and with 2 items prorated Category Missing Frequency table: WSD40P: 40-50% Prorate Count Cumulative Percent Cumulative Count Percent Since an error of ±1 represents 25% change in the scale score, this just might be significant given such a low score range in the first place. In reality, the WARS Self-Aggression scale is not a scale at all, but merely a collection of items. A quick item analysis of the 4 items comprising the WARS Self Aggression scale shows variable WARS112 WARS113 WARS114 WARS115 Summary for scale: Mean= Std.Dv.= Va Cronbach alpha: Standardized alpha: Average inter-item corr.: Mean if Var. if StDv. if Itm-Totl Alpha if deleted deleted deleted Correl. deleted Three features stand out the very low mean score (0.30 out of 4), a standard deviation over twice as large as the mean (0.79), and an alpha of just Whether we prorate or not, this scale is making poor measurement as a summated scale. Look at the frequency distribution of scores Charles Marley and Paul Barrett: The State Hospital, Carstairs 33
34 Category Missing Frequency table: WRSSATOT: WARS Self A Count Cumulative Percent Cumulative Count Percent What this shows is that out of 70 patients, only 5 score higher than 1.0 on the scale. The items of this scale are: 12. Talked of suicide 13. Attempted suicide 14. Talked of injuring self 15. Attempted to injure self. If we look at the frequency distributions for each item Category no yes Missing Frequency table: WARS112: talked suicide (w Count Cumulative Percent Cumulative Count Percent Category no yes Missing Frequency table: WARS113: talked self-injury Count Cumulative Percent Cumulative Count Percent Category no yes Missing Frequency table: WARS114: attempted self-i Count Cumulative Percent Cumulative Count Percent Category no yes Missing Frequency table: WARS115: attempted suicid Count Cumulative Percent Cumulative Count Percent I suspect that the best way of treating this scale is to actually use the individual items as indicators rather than try and summate them as though they measured a single psychological attribute. However, I digress! Prorating this scale seems a somewhat arbitrary action, for the meaning of items 12 and 13 is quite different from that in items 14 and 15. Prorating items 14 and 15 on the basis of 12 ad 13 would be a very strange assumption about suicide being equivalent to self-harm (a scale of items assumes unidimensionality of measurement). Prorating the scale even with one item exclusion may be unwarranted because of the item mix. I suspect that when we come to analysis of the WARS, an individual item profile analysis per patient may be the significant analysis, rather than relying upon summated scales. However, I suggest we continue with a single item-prorating as the most conservative way forward at the moment. Charles Marley and Paul Barrett: The State Hospital, Carstairs 34
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