﻿ User-friendly rescaling: zero point and unit

# User-friendly rescaling: zero point and unit

Winsteps sets the zero point (local origin) at the mean item difficulty (including extreme items if UEXTREME=Yes). The unit of measurement is the logit. Changing the zero point and transforming logits into other units can be done with UIMEAN=, UPMEAN=, USCALE=. These transformed measurements can be more meaningful for particular applications, see Chapter 8 of BTD. Anchor values are treated according to UASCALE=

Example 0: we observe the (current mean) and (current S.D.) of a set of measures. We want (desired mean) and (desired S.D.) then

new USCALE = (current USCALE=) *(desired S.D.) / (current S.D.)

new UIMEAN = desired mean - (current mean - current UIMEAN=)*(new USCALE=) / (current USCALE=)

we observe the (current high) and (current low). We want (desired high) and (desired low) then

new USCALE = (current USCALE=) *(desired  high-desired low)) / (current high-current low)

new UIMEAN = desired mean - (current mean - current UIMEAN=)*(new USCALE=) / (current USCALE=)

Example 1: CHIPs are a useful transformation, in which 1 logit = 4.55 CHIPs. In this user-scaling system, standard errors tend to be about 1 CHIP in size. The recommended control variable settings are:

USCALE = 4.55

UIMEAN = 50

UDECIM = 1

MRANGE = 50

The probability structure of this relationship is:

Probability of Success

Difference between Person Ability Measure and Item Difficulty Measure in CHIPs

.10 -10

.25 -5

.50 0

.75 5

.90 10

Example 2: WITs are one tenth the size of CHIPs, enabling the elimination of decimals from your output tables.

USCALE = 45.5

UIMEAN = 500

UDECIM = 0

MRANGE = 500

Example 3: You want the lowest reportable person measure to be 0 and the highest to be 100. Looking at Table 20, you see the extreme values are -4.53 and +5.72. You have not used USCALE= and UIMEAN=, so the current values are USCALE=1, UIMEAN=0.

Therefore,

"wanted low" person measure = 0

"wanted high" person measure = 100

"wanted range" = "wanted high" - "wanted low" = 100 - 0 = 100

"current low" person measure = -4.53

"current high" person measure = 5.72

"current range" = "current high" - "current low" = 5.72 - -4.53 = 10.25

USCALE= (wanted range) / (current range) = 100 / 10.25 = 9.76

UIMEAN= ( wanted low ) - ( current low * USCALE ) = 0 - ( -4.53 * 9.76 ) = 44.20

Required values are:

USCALE = 9.76

UIMEAN = 44.20

UDECIM = 0 to show no decimal places in report

Double checking, when previous values are UIMEAN=0, USCALE=1:

low value = (current low)*(USCALE=) + (UIMEAN=) = (-4.53 * 9.76) + 44.20 = -0.01 = 0

high value = (current high)*(USCALE=) + (UIMEAN=) = (5.72 * 9.76) + 44.20 = 100.02 - 100

Example 4:  You want the lowest reportable person measure to be 0 and the highest to be 100. Looking at Table 20, you see the extreme values are -4.53 and +5.72. The current values are USCALE=2, UIMEAN=1.5

Therefore,

"wanted low" person measure = 0

"wanted high" person measure = 100

"wanted range" = "wanted high" - "wanted low" = 100 - 0 = 100

"current low" person measure = -4.53

"current high" person measure = 5.72

"current range" = "current high" - "current low" = 5.72 - -4.53 = 10.25

"current UIMEAN" = 1.5

"current USCALE" = 2

USCALE= [(wanted range) / (current range)]/(current USCALE) = [100 / 10.25]/2 = 9.76/2 = 4.88

UIMEAN= ( wanted low ) - [(current low-current UIMEAN) * USCALE ] = 0 - [( -4.53-1.5) * 9.76] = 58.85

Required values are:

USCALE = 4.88

UIMEAN = 58.85

UDECIM = 0 to show no decimal places in report

Double checking is most easily done by looking at the new Table 20.

Example 5: You want the lowest reportable person measure to be 100 and the highest to be 900. Looking at Table 20, you see the extreme values are -4.53 and +5.72. Looking at the second page of output, you see the current values are USCALE=1 and UMEAN=0.

USCALE= (previous USCALE=) * (wanted range: 900 - 100) / (reported range: 5.72 - -4.53) = 1 * 800 / 10.25 = 78.05

UMEAN= (wanted low) - (reported low - previous UMEAN=)*(wanted range)/(reported range) = 100 - (-4.53 - 0)*800/10.25 = 453.56

UDECIM = 0 to show no decimal places in report

Example 6: You want norm-referenced user-scaling, such that the person mean is 0.0, and the person   standard deviation is 1.0.

In a standard analysis, set:

UDECIM=4

USCALE=1

UMEAN=0

Look at Table 18

+------------------------------

|ENTRY    RAW

|NUMBER  SCORE  COUNT  MEASURE

|------------------------------

| MEAN     6.7   14.0  -.3728

| P.SD     2.4     .0  2.2202

+------------------------------

Set (either in a new analysis, or using the "Specification" pull-down menu

USCALE = 1/person S.D. = 1/2.2202 = 0.4504

UMEAN = - person mean/person S.D. = - (-.3728)/2.2202 = 0.1679

Look at Table 18

+------------------------------

|ENTRY    RAW

|NUMBER  SCORE  COUNT  MEASURE

|------------------------------

| MEAN     6.7   14.0   .0000

| P.SD     2.4     .0  1.0000

+------------------------------

Example 7: You want to give your pass-fail point the value 400 and 100 to the lowest reported measure. Inspecting your output you see that the pass-fail point is at 1.5 logits, and that -3.3 logits is the lowest reported measure.

Then 400-100 new units = 1.5 - (-3.3) logits, so

USCALE = 300 / 4.8 = 62.5

UMEAN = 400 - (1.5) * 62.5 = 306.25

Then: 1.5 logits = 306.25 + 1.5*62.5 = 400

-3.3 logits = 306.25 - 3.3*62.5 = 100

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