5.5. Bias

Sources of bias discussed in this section cover specific measurement configurations. Measurements on test items are usually made on a single day, with a single operator, with a single instrument, etc. Even if the intent of the uncertainty is to characterize only those measurements made in one specific configuration, the uncertainty must account for any significant differences due to:
  1. instruments
  2. operators
  3. geometries
  4. other
Calibrated instruments do not normally fall in this class because uncertainties associated with the instrument's calibration are reported as type B evaluations, and the instruments in the laboratory should agree within the calibration uncertainties. Instruments whose responses are not directly calibrated to the defined unit are candidates for type A evaluations. This covers situations where the measurement is defined by a test procedure or standard practice using a specific instrument type.
If measurements for only one configuration are of interest, such as measurements made with a specific instrument, or if a smaller uncertainty is required, the differences among, say, instruments are treated as biases. The best strategy in this situation is to correct all measurements made with a specific instrument to the average for the instruments in the laboratory and compute a type A uncertainty for the correction. This strategy, of course, relies on the assumption that the instruments in the laboratory represent a random sample of all instruments of a specific type.
However suppose that it is only possible to make comparisons among, say, two instruments and neither is known to be 'unbiased'. This scenario requires a different strategy because the average will not necessarily give an unbiased result. The best strategy if there is a significant difference between the instruments, and this should be tested, is to apply a 'zero' correction and assess a type A uncertainty of the correction.
The discussion above is intended to point out that there are many possible scenarios for biases and that they should be treated on a case-by-case basis. A plan is needed for:

  • gathering data
  • testing for bias (graphically or statistically)
  • estimating biases
  • assessing uncertainties associated with significant biases.
Measurements needed for assessing biases among instruments, say, requires a random of sample of I (I > 1) instruments from those available and measurements on Q (Q >1) artifacts with each instrument. The same can be said for the other sources of possible bias. General strategies for dealing with significant biases are given in the table below.

Strategies for assessing corrections and uncertainties associated with significant biases

Type of bias
Examples
Type of correction
Uncertainty
1. Inconsistent
Sign change (+ to -) Varying magnitude
Zero
Based on maximum bias
2. Consistent
Instrument bias ~ same magnitude over many artifacts
Bias (for a single instrument) = difference from average over several instruments
Standard deviation of correction
3. Not correctable because of sparse data - consistent or inconsistent
Limited testing; e.g. only 2 instruments, operators, configurations, etc.
Zero
Standard deviation of correction
4. Not correctable - consistent
Lack of resolution, non-linearity, drift
material inhomogeneity
Zero
Based on maximum bias

5.5.1. Inconsistent bias

If there is significant bias but it changes direction over time, a zero correction is assumed and the standard deviation of the correction is reported as a type A uncertainty; namely,

The equation for estimating the standard deviation of the correction assumes that biases are uniformly distributed between {-max |bias|, + max |bias|}. This assumption is quite conservative. It gives a larger uncertainty than the assumption that the biases are normally distributed. If normality is a more reasonable assumption, substitute the number '3' for the 'square root of 3' in the equation above.
Example of inconsistent bias. The results of resistivity measurements with five probes on five silicon wafers are shown below for probe #283, which is the probe of interest at this level where the artifacts are 1 ohm.cm wafers. The bias for probe #283 is negative for run 1 and positive for run 2 with the runs separated by a two month time period. The correction is taken to be zero. 


          Table of biases (ohm.cm) for probe 283
            Wafer Probe    Run 1       Run 2

            -----------------------------------

              11   283   0.0000340  -0.0001841
              26   283  -0.0001000   0.0000861
              42   283   0.0000181   0.0000781
             131   283  -0.0000701   0.0001580
             208   283  -0.0000240   0.0001879

          Average  283  -0.0000284   0.0000652
A conservative assumption is that the bias could fall somewhere within the limits ± a where a = maximum bias or 0.0000652 ohm.cm. The standard deviation of the correction is included as a type A systematic component of the uncertainty.

5.5.2. Consistent bias

Bias which is significant and persists consistently over time for a specific instrument, operator, or configuration should be corrected if it can be reliably estimated from repeated measurements. This assumes the level of the bias is essentially the same for all materials of interest. Results are then corrected to:

Corrected result = Measurement - Estimate of bias

Given the measurements,

on Q artifacts with I instruments, the average bias for instrument, I' say, is

where

The correction that should be made to measurements made with instrument I' is

.

The type A uncertainty of the correction is the standard deviation of the average bias or

Example of consistent bias This example considers the case where measurements will be made with one instrument, and the reported values will be corrected for bias due to this instrument. The case where any one of the probes could be used to make measurements is treated as analysis of random error.

The table below shows resistivity measurements (ohm.cm) made with 5 probes on 5 silicon wafers where the average for each wafer is subtracted from each measurement. The differences, as shown, represent the biases for each probe with respect to the other probes. Probe 2362 has an average bias, over the five wafers, of -0.02724 ohm.cm. If measurements made with this probe are corrected for this bias, the standard deviation of the correction is a type A uncertainty. 


 Table of biases for probes and silicon wafers (ohm.cm)

                          Wafers
 Probe       138      139       140       141      142
---------------------------------------------------------
     1    0.02476  -0.00356   0.04002   0.03938   0.00620
   181    0.01076   0.03944   0.01871  -0.01072   0.03761
   182    0.01926   0.00574  -0.02008   0.02458  -0.00439
  2062   -0.01754  -0.03226  -0.01258  -0.02802  -0.00110
  2362   -0.03725  -0.00936  -0.02608  -0.02522  -0.03830

Average bias for probe #2362 = - 0.02724

Standard deviation of bias = 0.01171 with 4 degrees of freedom  

Standard deviation of correction = 0.01171/sqrt(5) = 0.00523
The graphs show differences from the average for each wafer plotted versus wafer identification with instruments individually identified by a special plotting symbol. The graphs confirm that probe 2362, (#5 on the graph) which is the instrument of interest for this measurement process, consistently reads low relative to the other probes. This behavior is consistent over 2 runs which are separated by a two month time period.


Run 1

Differences from the wafer average for each probe with probes coded:

(1 = #1; 2 = #281; 3 = #283; 4 = #2026; 5 = #2362).

The graph shows that probe #2362 is biased low relative to the other probes.

Run 2

Differences from the wafer average for each probe with probes coded:

(1 = #1; 2 = #281; 3 = #283; 4 = #2026; 5 = #2362).

The graph shows that probe #2362 is biased low relative to the other probes.
Because there is significant and consistent bias for probe 2362, measurements made with that instrument should be corrected for its average bias relative to the other instruments.

5.5.3. Bias with sparse data

The purpose of this discussion is to outline methods for dealing with biases that may be real but which cannot be estimated reliably because of the sparsity of the data. For example, a test between two, of many possible, configurations of the measurement process cannot produce a reliable enough estimate of bias to permit a correction, but it can reveal problems with the measurement process. The strategy for any significant bias is to apply a 'zero' correction. The type A uncertainty component is the standard deviation of the correction. For inconsistent bias, the standard deviation of the correction is taken to be

For consistent bias the standard deviation of the correction is taken to be

where the standard deviation in the equation is the standard deviation computed from the differences between the two configurations and N is the number of measurements in each configuration.

Example of bias from sparse data. An example is given of a study of wiring settings for a single gauge. The gauge, a 4-point probe for measuring resistivity of silicon wafers, can be wired in several ways. Because it was not possible to test all wiring configurations during the gauge study, measurements were made in only two configurations as a way of identifying possible problems.
Measurements were made on five wafers over six days (except for day 2 on wafer 39) with probe #2062 wired in two configurations. This sequence of measurements was repeated after about a month resulting in two runs. Differences between measurements in the two configurationson on the same day are shown below. 


Differences between wiring configurations

  Wafer  Day  Probe   Run 1    Run 2

   17.   1      2062.   -0.0108    0.0088
   17.   2      2062.   -0.0111    0.0062
   17.   3      2062.   -0.0062    0.0074
   17.   4      2062.    0.0020    0.0047
   17.   5      2062.    0.0018    0.0049
   17.   6      2062.    0.0002    0.0000

   39.   1      2062.   -0.0089    0.0075
   39.   3      2062.   -0.0040   -0.0016
   39.   4      2062.   -0.0022    0.0052
   39.   5      2062.   -0.0012    0.0085
   39.   6      2062.   -0.0034   -0.0018

   63.   1      2062.   -0.0016    0.0092
   63.   2      2062.   -0.0111    0.0040
   63.   3      2062.   -0.0059    0.0067
   63.   4      2062.   -0.0078    0.0016
   63.   5      2062.   -0.0007    0.0020
   63.   6      2062.    0.0006    0.0017

  103.   1      2062.   -0.0050    0.0076
  103.   2      2062.   -0.0140    0.0002
  103.   3      2062.   -0.0048    0.0025
  103.   4      2062.    0.0018    0.0045
  103.   5      2062.    0.0016   -0.0025
  103.   6      2062.    0.0044    0.0035

  125.   1      2062.   -0.0056    0.0099
  125.   2      2062.   -0.0155    0.0123
  125.   3      2062.   -0.0010    0.0042
  125.   4      2062.   -0.0014    0.0098
  125.   5      2062.    0.0003    0.0032
  125.   6      2062.   -0.0017    0.0115
A plot of the differences between the 2 configurations shows that the differences for run 1 are, for the most part, < zero, and the differences for run 2 are > zero.
The average and standard deviation computed from the N = 29 differences in each run from the table above are shown along with corresponding t-values which confirm that the differences are significant, but in opposite directions, for both runs. 

Average differences between wiring configurations
 Run Probe    Average    Std dev    N    t
    
  1    2062      - 0.00383       0.00514     29    - 4.0     
  2    2062      + 0.00489       0.00400     29    + 6.6

The bias is considered to be significant if

For this study, the type A uncertainty for wiring bias is

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