d2 Test of Attention - Revised (d2-R)

Abbreviations: d2-R, d2 Revision test

The d2-R manual gives no information on how the standard score norms were calculated from the raw scores. In part this can be inferred from the table values, but in part it remains unclear, as is presented below for each parameter individually.

The raw score range of E% goes from 0% (no errors, good performance) to (theoretically) 100%. In practice, raw scores between 0 and about 50 occur, with a mean close to 10, at the upper performance end of the distribution. E% thus yields skewed raw scores. The norming of the d2-R accounts for this by converting the raw scores into standard scores via percentile-to-normal transformations (percentiles) and thereby linearizing them. Figure 1 shows this transformation on the basis of the d2-R norm tables for four age groups, where the standard scores in the manual, with a mean of 100 and a standard deviation of 10, were converted into the IQ metric with a mean of 100 and a standard deviation of 15 used in TDB2Online, and smoothing with moving averages was also applied.

Figure 1: Transformation of the E% raw scores into IQ standard scores

The deviation of these curves from a straight line is easy to see. From the standard scores, the actual raw score distributions of E% can be back-calculated. Figure 2 shows the distributions for the four age groups from 15 years of age.

Figure 2: Raw score distributions of E% in four age groups

The skewness is apparent at first glance. It can also be seen that the age group of the 40–60-year-olds makes somewhat more errors on average than the other groups, while at the same time the distribution is somewhat narrower and extreme values occur less often.

CP theoretically has a raw score range from 0 to 308, even though the range occurring in practice is smaller. CP was introduced as the parameter for concentration performance as early as the 9th edition of the d2. Since the raw scores largely correspond to a normal distribution, CP was simply transformed linearly into standard scores there, based on the means and standard deviations of the respective age group. The d2-R manual provides no information on how the raw scores were transformed into standard scores. Figure 3 shows the graph of the transformation as it results from the tabulated values in the manual (with the IQ metric of TDB2Online instead of the SW metric of the d2-R). The steps that would result from the standard scores tabulated as whole numbers were smoothed with a third-order moving average.

Figure 3: Transformation of the CP raw scores into IQ standard scores

It can be seen that these are not simple linear transformations. However, they are not percentile transformations either. The graphs each contain (at least) two points at which the otherwise prevailing linearity is abandoned and the slope of the curve changes. It seems to have been important to those who constructed the norms (or resulted from the method used) that the graphs run linearly over long stretches. With a normal percentile transformation, this regularity would not exist.

In Figure 4, the graphs of the tabulated values for the two youngest age groups are contrasted with the straight lines that would result from the means and standard deviations given in Table 34 of the manual (shown dashed in the figure).

Figure 4: Transformation of the CP raw scores into IQ standard scores with superimposed linear transformation (dashed)

It can be seen that the curves of the same color deviate from each other, in the extreme ranges in part by up to one standard deviation. Evidently, then, an attempt was made to somehow take into account the deviation of the empirical distribution from a normal distribution. However, there is no information on how this was done or how well the transformation succeeded. The coarse and only stepwise correction shows that this could certainly have been done differently.

If the actual raw score distributions of CP are back-calculated from the tabulated standard scores, one arrives at the distributions in Figure 5.

Figure 5: Raw score distributions of CP in four age groups

As expected, the group of 20–39-year-olds shows the best performance and the group of 15–16-year-olds the worst. The expected positive correlation between mean and standard deviation is also evident: the higher the means, the broader the distribution.

PT, the speed measure of the d2-R, also theoretically has a raw score range from 0 to 308. The range occurring in practice is considerably smaller. In the old d2, GZ, the number of characters processed, is the measure of processing speed. In the 8th and 9th editions of the d2, GZ had in each case been transformed linearly into standard scores. Figure 6 shows the graph of the standard score transformation of PT as it results from the tabulated values in the manual (with the IQ metric of TDB2Online instead of the SW metric of the d2-R). The steps that would result from the standard scores tabulated as whole numbers were smoothed with a third-order moving average.

Figure 6: Transformation of the PT raw scores into IQ standard scores

It can be seen that these are almost linear transformations, but not quite. At the ends of the distributions, different deviations from linearity are visible. This becomes clearer in Figure 7, in which the graphs of the tabulated values for the two youngest age groups are contrasted with the graphs that would result from the means and standard deviations given in Table 34 of the manual (shown dashed).

Figure 7: Transformation of the PT raw scores into IQ standard scores with superimposed linear transformation (dashed)

In the 15–16-year-olds, the tabulated values deviate considerably from the simple linear values, especially in the extreme ranges; in the 16–17-year-olds, the deviation is hardly noticeable. For PT, too, an attempt was thus made to somehow take into account the deviation of the empirical distribution from a normal distribution, even though this is described nowhere and the result is not convincing.

If the actual raw score distributions of PT are back-calculated from the standard scores, one arrives at the distributions in Figure 8.

Figure 8: Raw score distributions of PT in four age groups

Here, too, the group of 20–39-year-olds shows the best performance and the group of 15–16-year-olds the worst. Differences in distribution shape are less conspicuous than for CP.

The figures in this section summarize the performance norms and age norms of the three parameters of the d2-R as they appear in TDB2Online.

In the adult range, norms in TDB2Online are presented in 5-year intervals where possible. This is not possible for the d2-R because the tabulated groups are too broad. To interpolate the values, one would need at least the mean ages of the groups, which the manual nowhere states.

The same applies to the performance scores, which in TDB2Online normally reflect the performance of young adults, mostly in the age range from 20 to 24 or from 20 to 29. For the d2-R, we have to take the entire group of 20–39-year-olds as the reference basis. Despite the broad age range, the group is still somewhat better on all parameters than the 17–19-year-olds, who would be the alternative.

Figure 9 shows, in a composite graphic, the relationship between raw and performance scores as well as the (average) influence of age on performance in the parameter percentage of errors (E%). In this and all following figures, the vertical bars mark the performance scores from 40 to 145. On the black horizontal line, the raw scores are plotted in their correct positions. The lines drawn alternately in red and blue contain the norm limits for all age groups in overview. The five marks on each line stand for the percentile ranks 2.5, 16, 50, 84, and 97.5. One such line, namely the one that corresponds to the age group of the examinee, is drawn (in a different form) in the TDB2Online profile sheet to help with the individual interpretation of the test results. Because the performance scores are calculated from the results of the 20–39-year-old young adults, the 5 marks of the age group 20–39 lie exactly on the performance scores 70, 85, 100, 115, and 130.

Figure 9: Performance scores for the parameter percentage of errors (E%, see text)

One can see, on the one hand, the nonlinear transformation of raw scores into performance scores. On the other hand, the lower standard deviation of the older adults is striking, which was already noticed above in the distribution analysis. Whether this is actually an age-associated property of the test or - as I rather suspect - results from differing sample characteristics of the age groups must remain open.

Figures 10 and 11 show the same graphic for the parameters Concentration performance (CP) and Processed targets (PT).

Figure 10: Performance scores for the parameter Concentration performance (CP, see text)

Figure 11: Performance scores for the parameter Processed targets (PT, see text)

For both parameters, the standard scores behave largely linearly with respect to the raw scores, and the ranges of values are also similar. The decline in performance scores from young adulthood (20–39 in the absence of narrower age groups) to somewhat later adulthood (40–60) amounts to barely a quarter of a standard deviation for the speed measure PT. For similar speed measures in other tests, the age-related loss in this age range is rather half a standard deviation. This, too, raises doubts about the adequacy of the sampling in this age group. It is probably too high for both parameters (accuracy and speed).

In addition to the d2-R presented here, TDB2Online can also score the 9th edition of the old d2. Since there is no separate test description for the old d2 in the test documentation of TDB2Online, the most important differences will at least be presented here in three areas: parameters, normative sample, calculation of the standard scores, and comparison of the norms in the last three versions.

For each of the three parameters of the d2-R, there were similar parameters in the earlier editions, although the differences are in individual cases quite relevant, even if they are not discussed further here.

Up to the 8th edition, GZ-F served as the parameter for overall performance, calculated from the total number of characters processed (GZ), reduced by the number of errors (F; sum of omission and confusion errors). In the 8th edition, CP was discussed for the first time as a replacement for GZ-F, after several articles criticizing GZ-F and proposing CP as an alternative had appeared in the early 1990s. In the 8th edition, however, only rough estimates for CP were given on the basis of the existing norms. The 9th edition contained new norms that listed CP as an index of concentration performance. GZ-F was still also included in the norms.

Instead of the processed targets (PT), the total number of characters processed (GZ) was calculated up to the 9th edition. Both are indices of processing speed.

For the percentage of errors (E%), up to the 9th edition the number of errors was divided by GZ, in the d2-R by PT. Both are accuracy indices.

The normative sample for the 8th edition (1994) was, in the adult range, still based entirely on the data collected at the Institute for Safety in Mining, Industry and Transport. At least these norms distinguished between persons in their fourth and fifth decades, which was no longer the case later.

For the 9th edition (2002), new norms were collected in 1999 and 2000 in four federal states. The adolescent age groups 15–16 and 17–19, which are of interest for TDB2Online, are well populated with 487 and 409 validly tested persons; this also applies to the young adults (20–39) with 731 persons, but no longer to the group of 40–60-year-olds with 293 persons. It is particularly regrettable that this group is not broken down more finely, since considerable age-associated performance losses must already be expected in this age range. Little can be said about the representativeness of the norms. The manual states that “a random sampling technique was applied”⁵⁾ . One of my favorite sentences is the remark somewhat further down in the text: “Cautiously put, this means that the goal of establishing norms that are as representative as possible can be considered approximately achieved.” In the absence of demographic characteristics of the sample, a comparison with the intended population is not possible.

For all three editions, the 8th, 9th, and 10th, it thus holds that although many persons were studied, representativeness of the samples was probably never really aimed for.

In the two earlier editions of the d2, the 8th and 9th, the standard scores for all parameters except E% were simply calculated linearly from the mean and standard deviation. This can be seen by looking at the parameter GZ-F, which is normed both in the last two editions of the d2 and in the d2-R.

In the 8th and 9th editions, the quantity scores (GZ, GZ-F, CP) were simply normed linearly using means and standard deviations. For the non-normally distributed E%, in the 8th edition only the raw scores at the 10th, 25th, 50th, 75th, and 90th percentiles were given; in the 9th edition, the 1st, 5th, 95th, and 99th percentiles were added. Only in the d2-R are the norms for E% also available in relatively fine gradation: for all T scores between 70 and 130, a raw score or a raw score range is given, evidently made possible by finer percentile calculations.

The d2-R manual also contains norms for the parameter GZ (old). This speed index of the old d2 can also be calculated from the d2-R test sheet, because the first 47 characters of each row correspond exactly to the old d2. (On page 39 of the d2-R manual, a theoretical conversion key is also given by which GZ can be calculated from PT. In fact, however, the GZ values calculated in this way are lower than those actually counted, as a comparison of the means of PT and GZ (old) in the norming section of the manual shows.)

The E% values are also quite well comparable if those obtained in the d2-R, which are based on PT, are multiplied by 2.2. This is the factor by which GZ is larger than PT. This yields the old E% value of the d2.

We first look at the GZ quantity scores. Figure 12 shows the transformation graphs of the GZ (old) raw score into IQ standard scores according to the d2-R norms, smoothed with moving averages.

Figure 12: Transformation of the GZ (old) raw scores into IQ standard scores

Figure 13 shows the corresponding raw score distributions for the four age groups.

Figure 13: Raw score distributions of GZ (old) in four age groups

In both figures (although it is more conspicuous in the distribution), one sees that there are clear ceiling effects in all age groups over 16 years. As a 25-year-old, one could not obtain an IQ-scaled standard score higher than 135 even with a maximum raw score of 658. The range of values, that is, the number of characters per row, was too small on the old d2 test sheet. This was corrected in the d2-R, in which each row contains 57 characters instead of the 47 in the d2.

Figure 14 contrasts, for the 20–39-year-old adults, the norms of the d2-R with those of the 8th and 9th editions of the d2. Plotted here are the standard scores originally used by the d2 and d2-R, which have a standard deviation of 10.

Figure 14: Comparison of the GZ norms of the 20–39-year-olds in the last three editions of the d2

For GZ, there are practically no relevant differences between the 9th edition of the d2 and the d2-R. The two curves run largely identically. Large differences, by contrast, occurred in the change from the 8th to the 9th edition. According to the norms of the 9th edition, an examinee's standard scores were suddenly at least one standard deviation lower than with the norms of the 8th edition. Unfortunately, such norm changes are hardly ever reported in the manual, although they can be significant in individual cases, especially with repeated measurements.

The situation is somewhat different for E%. Here, when comparing the norms, a correction factor must be taken into account, which incorporates on the one hand the different metric of PT and GZ and on the other hand the different number of rows. If this is done for the raw scores of the 8th and 9th editions, the results are those shown in Figure 15, in which the correspondingly corrected norms for adults between 20 and 39 years are shown for the three editions.

Figure 15: Comparison of the E% norms (with correction, see text) of the 20–39-year-olds in the last three editions of the d2

Again there are practically no relevant differences between the norms of the d2-R and those of the 9th edition of the d2, at least not in the middle range plus or minus two standard deviations. The norms for E% are available only roughly for the 9th edition and only very roughly for the 8th edition (in between, they were linearly interpolated for the figure). Between the 8th and 9th editions, a norm difference is again seen, which here, however, amounts to only about 1/4 to 1/3 of a standard deviation.