Multiple Regression via Measurement

The Complete Story on One Page

|-----+-----+-----+-----+-----+-----+-----+-----+-----|    BLOOD TEST
0   1 :    2       :     3    :(4):5:6: 7: 8  :   9   9   Urea Nitrogen
0    1   :   2   :  3  :  4 :(5) 6  :  7  : 8  : 9    9   Uric Acid
0    0    :        1     :   (2) : 3:45:67:8  : 9     9   Creatinine
 
0                   NO    :   (YES)                   1     GOUT?
|-----+-----+-----+-----+-----+-----+-----+-----+-----|
10   20    30    40    50    60    70    80    90   100  Measure
 
    1            1     3  5 65 87 31121 21 1        Patients with gout
 
                 1
       2  3   2  0   5 9  8 51 11 1                Patients without gout

Figure 1. Measure-based "multiple regression" of gout.

Kyle Perkins sent me physical dimensions, blood chemistries and diagnoses for 96 clinically-relevant patients, half of whom were thought to have gout. Since the precise relationship between the physical indicators and their medical implications is not known, I linearly transformed the original 3 physical and 5 chemical metrics into rating scales with ten categories, 0-9. The 5 diagnoses, reflecting expert (but imperfect) medical opinion, were coded as present=1, absent=0.

Rasch Multiple "Multiple Regression"

+----------------------------------------------------------------+
|  RAW                        |INFIT|OUTFIT|SCORE|               |
| SCORE  COUNT  MEASURE  ERROR|MNSQ |MNSQ  |CORR.|  ITEMS        |
|-----------------------------+-----+------+-----+---------------|
|   408     96    60.3A     .7| .83 | .82  |  .83| URIC ACID     |  INDEPENDENT
|   325     96    62.7A     .8| .70 | .75  |  .72| UREA NITROGEN |  VARIABLES
|   181     96    55.3A     .8| .89 |1.07  |  .62| CREATININE    |
------------------------------------------------------------------
|    48     96    53.7     1.9| .80 | .88  |  .61| GOUT          |  DEPENDENT
|    45     96    55.2     1.9| .91 |1.08  |  .51| HyperTense    |  VARIABLES
|    22     96    66.0     2.1|1.01 | .81  |  .39| Diuretic      |  Successful
------------------------------------------------------------------
|     6     96    79.5     3.5|1.19 |3.33  | -.03| KidneyStone   |  Unsuccessful
|     9     96    75.7     2.9|1.19 |4.78  | -.06| Diabetes      |
+----------------------------------------------------------------+

Table 1. Measure-based Rasch multiple "multiple regression"

Diagnoses "Regressed" on Blood Chemistry

------------------------------------------
|Opinion|  Count | Avg Meas | Diagnosis  |
------------------------------------------               DIAGNOSTIC
| absent|     48 |    47.70 | GOUT       |  DIAGNOSIS     MEASURES
|present|     48 |    59.44 |  1/0= +12  |   R= +.61        59
|----------------------------------------|
| absent|     51 |    48.93 | HYPERTENSE |  DIAGNOSIS               DEPENDENT
|present|     45 |    58.83 |  1/0= +10  |   R= +.51        59
|----------------------------------------|
| absent|     74 |    51.50 | DIURETIC   |  DIAGNOSIS               VARIABLES
|present|     22 |    60.52 |  1/0= +9   |   R= +.39        61
|-------+--------+----------+------------|
| absent|     90 |    53.65 | KidneyStone| No Diagnosis
|present|      6 |    52.42 |  1/0= -1   |   R= -.03
|----------------------------------------|
| absent|     87 |    53.76 | Diabetes   | No Diagnosis
|present|      9 |    51.77 |  1/0= -2   |   R= -.06
+----------------------------------------+

Table 2. Clinically-relevant diagnostic thresholds.

My next step was to discover which pieces of this heterogeneous collection would cooperate together to tell a meaningful story. From the necessarily rough ordinal data, I constructed Rasch measures.

Factor analysis of the residuals reported that blood creatinine, uric acid and urea nitrogen clustered with diagnoses of gout, hypertension and diuretic. Excluded were triglycerides, cholesterol, height, weight, surface area and the diagnoses of diabetes and kidney stones.

This led to the specification of a blood chemistry variable, based on levels of creatinine, uric acid and urea nitrogen, on which I could regress all five diagnoses. This was easily done by (1) calibrating these three items by themselves along with their rating scales; (2) anchoring these calibrations and the matching person measures, and (3) introducing gout and other diagnostic dichotomies into the analysis. [Later, Ben simplified (2) and (3) by using zero-item-weights for the non-measurement variables.]

The first results bear on the diagnosis of gout. Figure 1, "The Complete Story in One Picture", shows how well my new 3-blood-chemistry variable predicts a gout diagnosis. This Figure has the meaning of a conventional "multiple regression" in which gout is regressed on blood chemistry, but without the usual statistical obfuscation.

In Figure 1, there is the usual and unavoidable region of uncertainty, here between measures of 48 and 59. Otherwise the discrimination of gout/not gout is quite clear.

Investigation of the two "gout" patients with measures of 16 and 39 raise strong doubts about the accuracy of their recorded diagnoses. Neither has any blood chemistry evidence of gout.

Investigation of the two "no gout" patients with measures of 66 and 64 urge reconsideration of their diagnoses, since both have blood chemistries indicative of gout.

The measurement-based "regression" analysis laid out in Figure 1 was also done for the other four diagnoses. In fact, since the blood chemistry items, rating scales and person measures are all anchored, multiple "multiple regressions" can be done simultaneously.

These results are listed in Table 1, Rasch "Multiple Multiple Regression". The "multiple regression" correlations are simply the correlations between the scores on the diagnostic items and the patient measures. Gout=.61, Hypertension=.51, Diuretic=.39, Kidney Stone=-.03 and Diabetes=-.06 are clear enough. But much more accessible and useful are the diagnostic measure values and their explicit regions of uncertainty for the clinically-relevant patients summarized in Table 2.

Gout turns on at 59 with doubt down to 48.

Hypertension turns on at 59 with doubt down to 49.

Diuretic turns on at 61 with doubt down to 52.

Kidney Stones and Diabetes cannot be predicted from this blood chemistry variable.

Analyses like this could be run on all blood chemistry data for all diagnoses. Then relationships could be detected and variables constructed to implement all predictable diagnoses. The indicative levels for each diagnosis could be updated continually to focus on local and current practice and to keep pace with changing ways.

How simple, convenient, timely and useful!

Benjamin D. Wright with K. Perkins and K. Dorsey, Southern Illinois University

Multiple regression via measurement. Wright B.D. … Rasch Measurement Transactions, 2000, 14:1 p.729


Please help with Standard Dataset 4: Andrich Rating Scale Model



Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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