## 8. LINKING FUNCTION FOR INTERNATIONAL MARKER LEVELS

In addition to reporting results in terms of means, the TIMSS reports also portray the performance of students in each country in terms of the percentages of students exceeding each of three marker levels. Since the TIMSS assessments do not have any prespecified performance standards, three marker levels were chosen on the basis of the combined performance of all students participating in the TIMSS. These marker levels corresponded to the 90th, 75th, and 50th percentiles of the combined distribution of proficiency across all participating countries for a given subject and grade. These marker levels are named, respectively, the Top 10 Percent, the Top Quarter, and the Top Half, and are given in Table 13.

Table 13.—International marker levels of achievement for grade 8

 Subject Top 10 Percent Top Quarter Top Half Mathematics 656 587 509 Science 655 592 522

The predicted proportion of individuals in a given state that would exceed any given international marker level comes from determining a predicted value on the NAEP scale corresponding to the cutpoint on the TIMSS scale and then computing the proportion of students in that state whose NAEP plausible values exceed that NAEP cutpoint.

That is, one begins by linking TIMSS to NAEP. The equation for this linking is easily obtained by inverting Equation (1). Let y be a specified marker-level cutpoint, and let be the predicted cutpoint on the NAEP scale. The connection between y and is

(17)

where

The variance of is computed in a manner exactly analogous to that of the variance of . In fact, the equations for the variance can be determined from the previous derivations by making the following substitutions:

• and and y for x
• and for and and vice versa

The sole exception to this rule is that the marker level cutpoint, y, is taken in the TIMSS reports as a fixed value so that Var(y) has been taken to be 0. Following this convention, the variance of is

(18)

where the terms in Equation (18) include the components of variance due to sampling, measurement error, and temporal shift.

Table 14 gives the values of the predicted NAEP cutpoint, , and its standard error, for the three marker levels. Since these results are to be used for the reporting of public school state results, the linking was accomplished using the public school data.

Table 14.—Predicted NAEP cutpoints and their standard errors corresponding to the TIMSS marker levels—public school linking for grade 8

 Subject Top 10 Percent Top Quarter Top Half SE SE SE Mathematics 333.9 2.7 306.2 2.4 274.9 2.2 Science 188.2 2.1 167.8 1.9 145.2 1.8

Observe that, although the international marker levels are similar for grade 8 mathematics and science, the predicted NAEP cutpoints differ significantly. This is because of the differences in the metrics used in the NAEP and TIMSS scales. The TIMSS scales for each of the subjects were set to have a mean of 500 and a standard deviation of 100 across the participating countries. Conversely, the metric for the NAEP mathematics scale is a 500-point scale across the grades 4, 8, and 12 while the grade 8 science scale is expressed on a 300-point, within-grade metric.

Confidence intervals for the predicted proportion of students above a particular marker level are obtained by first identifying the lower and upper bound of a confidence interval on where SE() is the standard error of . For each of these cutpoints, one computes , the estimates of the proportion of students in a given state exceeding the two cutpoints. Each of these estimated proportions is accompanied by a standard error, which accounts for the effects of sampling and imprecision of measurement in estimating that proportion for the given state.

Since is less than and since the proportion, exceeding the lower cutpoint cannot be smaller than the proportion, exceeding the higher cutpoint, a conservative confidence interval about the predicted proportion in the state exceeding the marker level is

(19)