Correlations: point-biserial, point-measure, residual

In Rasch analysis, we use item correlations as an immediate check that the response-level scoring makes sense. If the observed correlation is negative, something may have gone wrong (MCQ miskey, reversed survey item, etc.)

In general, correlations are much too difficult to interpret, so we switch over to using mean-squares. The "expected correlation" indicates when conventional rules such as eliminate items with point-biserials less than 0.2 are misleading.

Item correlations are difficult to interpret because they are influenced by:

1. predictability of the data

2. targeting of the item on the person sample

3. distribution of the person sample

In Rasch analysis, we are chiefly concerned about the predictability of the data when assessing item quality, so we examine the predictability directly using the mean-square statistics, rather than indirectly through the correlations.

 

All correlations are computed as Pearson product-moment correlation coefficients. If you wish to compute other correlations, the required data are in XFILE= IPMATRIX=, IFILE= or PFILE=. The Biserial correlation can be computed from the point-biserial.

 

 

Point-correlations are always reported for items. Point-correlations are reported for persons when (i) all the items are dichotomies, (ii) all the items have three categories, or (iii) all the items are in the same item group (Andrich Rating Scale Model). Otherwise, the observed and expected correlations are reported as zero in PFILE=.

 

Example: Here are the point-biserial-type coefficient options for Example0.txt, Item 1 (Watch Birds).

 

In Table 14.1 and other measure Tables: Point-Biserial Correlations for dichotomies, and Point-Polyserial for polytomies

 

when PTBISERIAL=Yes

PTBSE is the point-biserial correlation between the responses to this item by each person and the total marginal score by each person (omitting the response to this item). This is the "point-biserial corrected for spuriousness". Henrysson, S. (1963). Correction for item-total correlations in item analysis. Psychometrika, 28, 211-218.

 

when PTBISERIAL=All

PTBSA is the point-biserial correlation between the responses to this item by each person and the total marginal score by each person (including the response to this item). This is the conventional point-biserial.

 

In Table 14.3 and other or distractor Tables:

when PTBISERIAL=Yes or PTBISERIAL=All

PTBSD is the distractor point-biserial correlation between the indicated response to this item (scored 1 and other responses scored 0) by each person and the total marginal score by each person.

 

There is a closer match between Table 14.1 and Table 14.3 when PTBISERIAL=All

 

PTBIS=Y or E (indicated by PTBSE): The point-biserial correlation rpbis for item i (when i=1,L for persons n=1,N) is the correlation between the observation for each person on item i and the total score for each person on all the items excluding item i (and similarly for the point-biserial for each person):

 

 

PTBIS=All (indicated by PTBSA): All the observations are included in the total score:

 

 

where X1,..,XN are the responses, and Y1,..,YN are the total scores. The range of the correlation is -1 to +1.

 

Under classical (raw-score) test theory conventions, point-biserial correlations should be 0.3, 0.4 or better. Under Rasch conditions, point-biserial (or point-measure) correlations should be positive, so that the item-level scoring accords with the latent variable, but the size of a positive correlation is of less importance than the fit of the responses to the Rasch model, indicated by the mean-square fit statistics.

 

Point-Measure Correlations

 

PTBIS=No (indicated by PTMEA): The correlation between the observations and the Rasch measures:

 

 

where X1,..,XN are the responses by the persons (or on the items), and Y1,..,YN are the person measures (or the item easinesses = - item difficulties). The range of the correlation is -1 to +1.

 

Jaspen, N. (1946) "Serial Correlation", Psychometrika, 11, 23-30.

Olsson, U., Drasgow, F. and Dorans, N.J. (1982) "The Polyserial Correlation Coefficient", Psychometrika, 47, 337-347.

 

---

 

The Expected Value of a Correlation

 

Interpreting an observed value is made easier if we can compare it with its expected value. Is is much higher than expected or much lower than expected?

 

The general formula for a Pearson correlation coefficient is shown above. Suppose that Xn is Xni the observation of person n on item i. Yn is Bn the ability of person n, then the point-measure correlation is:

 

 

According to the Rasch model, the expected value of Xni is Eni and its model variance around the expectation is Wni. For dichotomies, Eni is the Rasch probability of success and Wni = Eni(1-Eni). For polytomies, Eni and Wni are given by RSA, p. 100 also at www.rasch.org/rmt/rmt34e.htm.  For JMLE estimates, Sum(Eni) = Sum(Xni) for n=1,N. Thus an estimate of the expected value of the point-measure correlation is given by the Rasch model proposition that: Xni = Eni ±√Wni. Other variance terms are much smaller.

 

 

±√Wni is a random residual with mean 0 and variance Wni. Its cross-product with any other variable is modeled to be zero. Thus, simplifying,

 

and similarly for the point-biserial correlations. Here is an example point-measure correlation and its expected value computed with an Excel spreadsheet:

 

Table courtesy of Peter Karaffa

 

---

 

Point-Correlations and the Partial Credit Model or Grouped Rating Scale Model: ISGROUPS=

The usual computations for correlation coefficients assume that all scores or measures within each of the two variables is in the same metric. For instance, when correlating age with height for children, the ages are all in months and the heights are all in centimeters. If we mix ages in months with ages in years, or height in centimeters with height in inches, then the age-height correlations can become meaningless.

The same situation can arise when we use the Partial Credit Model (ISGROUPS=0) or Grouped Rating Scale Model (ISGROUPS=AABB..). In particular, the Expected value of a Point-Correlation can be reported as negative. This is a warning is that the reported correlations are meaningless.

 

---

 

Disattenuated correlation coefficients

"Attenuated" means "reduced". "Disattenuation" means "remove the attenuation".

The observed correlation between two variables is attenuated (reduced toward zero) because the variables are measured with error. So, when we remove the measurement error (by a statistical operation), the resulting correlation is disattenuated. Disattenuated correlations are always further from zero.

Algebraically:

{A} and {B} are the "true" values of two variables. Their true (disattenuated) correlation is

Disattenuated ("true") correlation = r(A,B)

But the observed values of the variables are measured with error {A±a}, {B±b}, so the observed correlation is

observed correlation = r(A,B) * √(var(A)*var(B))/√((var( + a +a ²)*(var( + b + b ²)).

"Disattenuation" reverses this process.

 

If the reliability of {A} is RA, and the reliability of {B} is RB, then the disattenuated correlation between {A} and {B} is:

disattenuated r(A,B) = r(A,B) / √(RA*RB).

 

---

 

Point-Correlations in Table 14.3, etc. and DISFILE= and IFILE=

 

For the item point-biserial correlations, PTBISERIAL=A and PTBISERIAL=E:,  scored observations are correlated with their corresponding person raw scores.For the point-measure correlations, PTBISERIAL=M, scored observations are correlated with their corresponding Rasch measures.

 

Each response code (scored or "missing") is correlated only with scored response codes. With PTBISERIAL=Yes or All, the correlation is between the occurrence and the person raw score, indicated by PTBSD CORR. When this correlation is high positive for a correct MCQ option, then the item exhibits convergent validity. When this correlation is low or negative for incorrect MCQ options, then the item exhibits discriminant validity. Krus, D. J. & Ney, R. G. (1978) Convergent and discriminant validity in item analysis. Educational and Psychological Measurement, 38, 135-137.
Each scored response code is correlated only with the other scored codes: "1" for target code "0" for other scored codes. . Missing responses are correlated with all responses.
PTBSA CORR is PTBIS = All (include the current item in the person score)
PTBSE CORR is PTBIS = Yes (exclude the current item in the person score)
PTMEA CORR is PTBIS = No (correlate with the person measure)
The "Missing" point-measure correlation is the correlation between the raw scores for the persons and the responses to the item with "missing" scored 1 and the other responses scored 0.
The category (or option) correlation is always computed the same way:
1. We focus on a particular response to a particular item.
2. Everyone who chose that response to the item is scored "1", everyone else is scored "0". Missing data are ignored.
3. The scores of "1" or "0" are correlated with the person measures.
4. We do this for every response to every item.

 

Example: Point-Measure Correlations of Person Measures with Item 6 Options a,b,c,d, with b scored 1, and missing data, x.

 

For the options reported in Table 14.3 and DISFILE=

 For each item and each option, a correlation is computed for each code in CODES=

   If the response in the data file is the code, then a value of 1 is assigned

   If the response in the data file is not the code and in CODES=, then a value of 0  is assigned

   If the response in the data file is not in CODES= and scored with MISSING-SCORED=, then a value of 0 is assigned.

   If the response in the data file is not in CODES= and not scored with MISSING-SCORED=, then the response is ignored.

   For each value of 0 or 1, an average response score is computed = (raw score / count scored observations)

   The correlation is between the (weighted) values and the average response scores or Rasch measures

     

 For each item, one correlation is computed for all codes in the data not in CODES=

   If the response in the data file is not in CODES=, then a value of 1 is assigned

   If the response in the data file is in CODES=, then a value of 0  is assigned

   For each value of 0 or 1, an average response score is computed = (raw score / count scored observations)

   The correlation is between the weighted values and the average response scores or Rasch measures

 

Different correlation values in DISFILE= vs. Table 14.3, etc. and IFILE= for apparently the same correlation:

 

If your data file contains unscored data or missing data, then the item point-correlations will differ from the option point-correlations. If you want the correlations to be the same, then force missing data to be scored 0:

1. Do your usual analysis

2. Output IFILE=if.txt, PFILE=pf.txt, SFILE=sf.txt

3. Add to your control file:

   IAFILE=if.txt  ; anchor everything at their estimated values

   PAFILE=pf.txt

   SAFILE=sf.txt

   CODES=your codes + all other codes in the data file. For example, if your codes are A,B,C,D and missing data is blank, then CODES=" ABCD "

4. Do the analysis again. Output Table 14.3, DISFILE=. The values should now agree.

5. Do NOT use the fit statistics estimated from this analysis.

 

Suggestion: since the point-biserial is not defined when there is unscored missing data, please use the point-measure correlation, PTBISERIAL=M.

 

---

 

Statistical Significance of a Correlation

 

Null hypothesis is that the correlation is 0.00.

 

Student's significance test:  t = r / sqrt[ (1—r2) / (N—2) ] with N-2 d.f. where N is the count of correlated cases.

 

Null hypothesis is that the correlation is another value, not 1 or -1

 

1. Use Fisher z' transformation to linearize hypothesized correlation and observed correlation:

z' = .5[ln(1+r) - ln(1-r)]

 

2. normal z = ( z'(observed) - z'(hypothesized) ) * sqrt (N-3)

 

Null hypothesis is that the correlation is 1 or -1

 

Use a number that means the same thing: 0.99 or -0.99