Standardized tests, whether to evaluate student performance in coursework or to choose among applicants for college admission or to license candidates for various professions, are often marathons. Tests designed to evaluate knowledge of coursework typically use the canonical hour; admissions tests are usually two to three hours; and licensing exams can take days. Why are they as long as they are? To answer this question, we must consider the purposes of the test. Most serious tests have serious purposes – admission to a college or not, getting a job or not, being allowed to practice your profession or not. The extent to which a test score can serve these purposes is its validity, which is usually defined as “the degree to which evidence and theory support the intended interpretations of those test scores.”Footnote ¹ But the validity of a test’s scores is bounded by the test’s reliability.Footnote ² Reliability is merely a standardized measure of the score’s stability, ranging from a low of 0 (essentially a random number) to a high of 1 (the score does not fluctuate at all). A test score that has low reliability must perforce have an even lower validity, and its usefulness diminishes apace.

Thus, the first answer to the question “Why are tests so long?” that jumps immediately to mind is derived from the inexorable relationship between a test’s length and its reliability. However, even though a test score always gets more reliable as the test generating it gets longer, ceteris paribus, the law of diminishing returns sets in very quickly. In Figure 1.1, we show the reliability of a typical professionally prepared test as a function of its length. The figure shows that the marginal gain of moving from a 30-item test to a 60- or even 90-item one is not worth the trouble unless such small additional increments in reliability are of practical importance. We must also note that reliability does drop precipitously as test length shrinks from 30 toward zero. But the puzzle still remains: Why are typical tests so long if, after a test’s length surpasses 30, it is about as reliable as we are likely to need. So, if such extra precision is rarely necessary, why are tests as long as they are?

Figure 1.1 Spearman–Brown function showing the reliability of a test as a function of its length, if a one-item test has a reliability of 0.15.

1.1 A Clarifying Example: The US Census

Our intuitions can be clarified with an example, the decennial US census. According to the 2020 decennial census, the United States had 331,449,281 residents; however, the Bureau of the Census estimates that this number has a nontrivial error of uncertain size. The estimate is that 782,000 more people resided in the United States than were reported, but it is also estimated that about one third of the time the estimate would be that the decennial census overcounted the population by at least 30,000 or undercounted the population by no less than 1,620,000 people. The budget of the 2020 census was $14.2 billion, or approximately $42.84 per person counted. Is it worth this amount of money to just get this single number? Before answering this question, consider the function shown in Figure 1.2, which provides the results of all decennial censuses for the past 150 years. The curve shown is a fitted quadratic function to the data from 1870 through 2010. The large dot associated with 2020 is the actual population estimate from the 2020 census; the value of the curve, which passes slightly above it, is the estimate obtained from this fitted function – 331,943,913. The difference between the two is 494,632 people, or 0.1%, an error certainly comparable to the estimated error from the decennial census. Obtaining this estimate cost only about an hour of a statistician’s time – totaling perhaps a couple of hundred dollars.

Figure 1.2 US population from 1870 to 2020.

Fitting a connecting function, like the quadratic shown in Figure 1.2, can have many uses. By far, the most important purpose is to provide, at a glance, an accurate representation of more than a century’s growth of a nation’s population – Henry D. Hubbard (in the preface to Brinton, Reference Brinton1939) memorably characterized this use when he pointed out the following: “There is a magic in graphs. The profile of a curve reveals in a flash a whole situation – the life history of an epidemic, a panic, or an era of prosperity. The curve informs the mind, awakens the imagination, convinces.”

Although the fitted curve provides only an approximation of the population between censuses, we can, by adopting very credible regularity assumptions, confidently use the curve to interpolate between censuses and obtain accurate estimates for any time in the 150-year range of the data.

A third use, and the one that we have illustrated here, is extrapolation 10 years beyond the 2010 census. Extrapolation, like interpolation, relies on regularity assumptions, but those assumptions become more heroic the further the estimate is from the data. As we have seen, predicting the 2020 US population from prior census results ending in 2010 yielded an estimate that is likely accurate enough for most applications. Were we to use the same function to predict further into the future, we would be less sure, and our uncertainty would, naturally, expand with the size of the extrapolation. Of course, there are more data-rich methods that could improve the accuracy of such extrapolations by making their inevitable underlying assumptions more credible.Footnote ³

And so, returning to the original question, is it worth $14.2 billion to just estimate this single number when it could have been determined as accurately in just a few minutes and be paid for out of petty cash? It doesn’t take a congressional study group or the Office of Management and the Budget to tell us that the answer is no. If all the census gave us was that single number, it would be a colossal waste of census workers’ time and taxpayers’ money. However, the Constitution and acts of Congress require that the census enumerate the resident population and report on population sizes for very small geographical regions, including for 2020 about 11,000,000 census blocks and 73,000 census tracts. These population values for small areas are needed for apportionment of the House of Representatives and for apportionment of state legislatures and other governmental units. They, together with survey data from the American Community Survey, are also valuable for allocation of social services, among other uses. The census provides such small area estimates admirably well, but to be able to do so requires massive data collection and so incurs a huge expense. Yet the importance of providing accurate answers to many crucially important small area questions makes its impressive cost unavoidable.

There are the two key lessons we should take from this census example:

1. 1. Obtaining an accurate estimate of the grand total is easy and cheap.

2. 2. Obtaining accurate small area estimates is hard and expensive, and hence should not be attempted unless such small area estimates are important enough to justify the vast increases in the resources of time and treasure that are required.

1.2 Back to Tests

Now let us return to tests. Instead of characterizing cost in terms of dollars (a worthwhile metric, for sure, but grist for another mill), let us instead use examinee time. Is it worth using an hour (or two or even more) of examinee time to estimate just a single number – a single score? Is the small marginal increase in accuracy obtained from a 60- or 90-item test over, say, a 30-item test worth doubling or tripling examinee time?

A glance at the gradual slope of the Spearman–Brown curve shown in Figure 1.1 as it nears its asymptote tells us that we aren’t getting much of a return on our investment. And multiplying the extra hour spent by each examinee by the millions of examinees that often take such tests makes this conclusion stronger still. What would be the circumstances in which a test score with a reliability of 0.89 will not suffice, but one of 0.91 would? Off hand, it is hard to think of any.

But, returning to the lessons taught us by census, perhaps there are other uses for the information gathered by the test that require additional length – the equivalent of the small area estimates of census. In testing, such estimates are usually called subscores – small area estimates on aspects of the subject matter of the test. On a high school math test, these might be subscores on algebra, arithmetic, geometry, and trigonometry. For a licensing exam in veterinary medicine, there might be subscores on the pulmonary system, the skeletal system, the renal system, and so on. There is even the possibility of cross-classified subscores; perhaps one on dogs; another on cats; and others on cows, horses, and pigs. Such cross-classified subscores are akin to the census having estimates by ethnic group and also by geographic location.

Thus, the production of meaningful subscores needed for important purposes would be a justification for tests that contain more items than would be required merely for an accurate enough estimate of total score. What is a meaningful subscore? It is one that is reliable enough for its prospective use and one that contains information that is not adequately focused (or is overly diluted) in the total test score.

There are at least three prospective uses of such subscores:

1. (i) To provide institutions with information used for major decisions such as admission, licensure, and immigration qualification

2. (ii) To aid examinees in assessing their strengths and weaknesses, often with an eye toward remediating the latter

3. (iii) To aid individuals and institutions (e.g., teachers and schools) in assessing the effectiveness of their instruction, again with an eye toward remediating weaknesses

In the first case, subscores need to be highly reliable, given the life-changing decisions dependent upon them. The demands on reliability increase when subscores must exceed a fixed value and when multiple subscores are involved.

In the second case, helping examinees, the subscores need to be reliable enough so that attempts to remediate weaknesses do not become just the futile pursuit of noise. And, obviously, the subscores must contain information that is focused on performance on the specific topic of interest and is not diluted over the broad range of topics contained in the total test score. We might call these two characteristics of a worthwhile subscore reliability and specificity. But for a subscore to have a specific focus apart from the total score, its information must be somewhat orthogonal to the total score; hence we shall designate this characteristic of a useful subscore orthogonality. Shortly, we will provide an introductory discussion of each of these two important characteristics and then tell the full story in Chapter 3. But first, let us drop back in time and see where the concept and use of subscores came from.

1.3 A Brief Account of the Long History of Tests and Subtests, Scores and Subscores

The use of mental tests appears to be almost as ancient as Western civilization. The Hebrew Bible (in Judges 12:4–6) provides an early reference to testing in Western culture.Footnote ⁴ It describes a short verbal test that the Gileadites used to uncover the fleeing Ephraimites hiding in their midst. The test was one item long. Candidates had to pronounce the word שיבולת (transliterated as shibboleth). Ephraimites apparently pronounced the initial sh as s. The consequences of this test were quite severe (the banks of the Jordan were strewn with the bodies of the 42,000 who failed). Obviously, any test that consists of but a single item can have no subscores. But there were much earlier tests that were longer and had subscores.

In his 1970 History, Philip DuBois reported that tests had been around for millennia, and whenever they consisted of more than a single item, the appeal of computing subscores has been irresistible.

There was some rudimentary proficiency testing that took place in China around 2200 bce, which predated the biblical testing program by almost a thousand years! The emperor of China is said to have examined his officials every third year. This set a precedent for periodic exams in China that was to persist for a very long time. In 1115 bce, at the beginning of the Chan dynasty, formal testing procedures were instituted for candidates for office. Job sample subtests were used, with proficiency required in (1) archery, (2) arithmetic, (3) horsemanship, (4) music, (5) writing, and (6) skill in the rites and ceremonies of public and social life. The Chinese discovered the fundamental truth that underlies the validity of testing – that a relatively small sample of an individual’s performance, measured under carefully controlled conditions, can yield an accurate picture of that individual’s ability to perform under much broader conditions for a longer period of time. The procedures developed by the Chinese are reasonably similar to the canons of good testing practice used today. For example, they required objectivity – candidates’ names were concealed to ensure anonymity; they sometimes went so far as to have the answers redrafted by another individual to hide the handwriting. Tests were often read by two independent examiners, with a third brought in to adjudicate differences. Test conditions were as uniform as could be managed – proctors watched over the exams given in special examination halls that were large permanent structures consisting of hundreds of small cells. The testing process was so rigorous that sometimes candidates died during the course of the exams.

This testing program was augmented and modified through the years and has been praised by many Western scholars. Voltaire and Quesnay advocated its use in France, where it was adopted in 1791, only to be (temporarily) abolished by Napoleon. It was cited by British reformers as their model for the system set up in 1833 to select trainees for the Indian civil service – the precursor to the British civil service. The success of the British system influenced Senator Charles Sumner of Massachusetts and Representative Thomas Jenckes of Rhode Island in the development of the examination system they introduced into Congress in 1868. This eventually led to George Hunt Pendleton proposing the eponymously entitled US Civil Service Act in January 1883.

The US military has arguably one of the most widely used and consequential testing programs in the United States, in terms of both number of examinees and length of time it has been in use. It also has been carefully thought through and researched over the better part of a century. Few testing programs can match the careful seriousness of its construction and use. We feel this makes it a worthy and informative illustration of the use of tests and subscores in support of evidence-based personnel decision-making. In the following narrative, we first trace the history of military testing in the United States, then move to a discussion of decisions based on total test scores and the use of subscores, and then conclude with an evaluation of the success of this approach as a guide to others who would like to use both test scores and subscores to generate evidence supporting claims about individuals and groups.

1.4 The Origins of Mental Testing in the US Military

During World War I, Robert M. Yerkes, president of the American Psychological Association, took the lead in involving psychologists in the war effort. One major contribution was the implementation of a program for the psychological examination of recruits. Yerkes formed a committee for this purpose that met in May 1917 at the Vineland Training School in New Jersey. His committee debated the relative merits of very brief individual tests versus longer group tests. For reasons of objectivity, uniformity, and reliability, they decided to develop a group test of intelligence.

The criteria they adopted (described in detail on page 62 of Philip DuBois’ Reference DuBois1970 book on the history of testing) for the development of the new group test were:

1. 1) Adaptability for group use

2. 2) Correlation with measures of intelligence known to be valid

3. 3) Measurement of a wide range of ability

4. 4) Objectivity of scoring, preferably by stencils

5. 5) Rapidity of scoring

6. 6) Possibility of many alternate forms so as to discourage coaching

7. 7) Unfavorableness to malingering

8. 8) Unfavorableness to cheating

9. 9) Independence of school training

10. 10) Minimum of writing in making responses

11. 11) Material intrinsically interesting

12. 12) Economy of time

In just 7 working days, they constructed 10 subtests with enough items for 10 different forms. They then prepared one form for printing and experimental administration. The pilot testing was done on fewer than 500 subjects. These subjects were broadly sampled and came from such diverse sources as a school for those with intellectual disabilities, a psychopathic hospital, a reformatory, some aviation recruits, some men in an officers’ training camp, 60 high school students, and 114 marines at a navy yard. They also administered either the Stanford–Binet intelligence test or an abbreviated form of it. They found that the scores of their test correlated 0.9 with those of the Stanford–Binet and 0.8 with the abbreviated Binet.

The items and instructions were then edited, time limits were revised, and scoring formulas were developed to maximize the correlation of the total score with the Binet. Items within each subtest were ordered by difficulty, and four alternate forms were prepared for mass administration.

By August, statistical workers under E. L. Thorndike’s direction had analyzed the results of the revised test after it had been administered to 3,129 soldiers and 372 inmates of institutions for mental defectives. The results prompted Thorndike to call this the “best group test ever devised.” It yielded good distributions of scores, and it correlated about 0.7 with schooling and 0.5 with ratings by superior officers. This test was dubbed Examination a.

In December of the same year, Examination a was revised once again. It became the famous Army Alpha. This version had only eight subtests; two of the original ones were dropped because of low correlation with other measures and because they were of inappropriate difficulty. The resulting test (whose components are shown below) bears a remarkable similarity to the structure of the modern Armed Services Vocational Aptitude Battery (ASVAB).

This testing program, which remained under Yerkes’ supervision, tested almost 2 million men. Two thirds of them received the Army Alpha, and the remainder were tested with an alternative form, Army Beta, a nonverbal form devised for illiterate and non-English-speaking recruits. Together they represented the first large-scale use of intelligence testing.

The success of the Army Alpha led to the development of a variety of special tests. In 1919, Henry Link discovered that a card-sorting test aided in the successful selection of shell inspectors and that a tapping test was valid for gaugers. He pointed out that a job analysis coupled with an experimental administration of tests thought to require the same abilities as the job and a validity study that correlated test performance with later job success yielded instruments that could distinguish between job applicants who were good risks and those who were not. L. L. Thurstone developed a “rhythm test” that accurately predicted future telegraphers’ speed.

Testing programs within the military became much more extensive during World War II. In 1939, the Personnel Testing Service was established in the Office of the Adjutant General of the Army. This gave rise to the Army General Classification Test (AGCT) that was an updated version of the Army Alpha. The chairman of the committee that oversaw the development of the AGCT was Walter V. Bingham, who served on the 1917 committee that developed Alpha. This test eventually developed into a four-part exam consisting of tests of (1) reading and vocabulary, (2) arithmetic computation, (3) arithmetic reasoning, and (4) spatial relations. Supplemental tests for mechanical and clerical aptitude, code learning ability, and oral trade were also developed. By the end of the war, more than 9 million men had taken the AGCT in one form or another. The navy and the army air forces participated in the same program but with some different tests that they required for their own purposes.

In 1950, the Armed Forces Classification Test was instituted to be used as a screening instrument for all services. It was designed to ensure appropriate allocation of talent to all branches. This was the precursor of the Armed Forces Qualification Test (AFQT) that led in turn to the Armed Services Vocational Aptitude Battery (ASVAB).

1.5 Subscore Reliability

Subscores’ reliability is governed by the same inexorable rules of reliability as overall scores – as a test’s length decreases, so too does its reliability. Thus, if we need reliable subscores, we must have enough items for that purpose. A glance at the low end of the curve in Figure 1.1 shows clearly that the marginal value of each additional item to a score’s reliability is much greater when there are few items than when there are many (the right side of the curve in Figure 1.1). But this means that to have reliable subscores, the overall length of a test would have to be greater than would be necessary for merely a single reliable total score.

For the second use, helping institutions, the test’s length might not have to increase, for the reliability would be calculated over the number of individuals from that institution who took the items of interest. If that number was large enough, the estimate could achieve high reliability.

And so it would seem that one key justification for what appears at first to be the excessive lengths of most common tests is to provide feedback to examinees in subscores calculated from subsets of the tests. Certainly, that is what was conveyed by the military’s guidance to examinees, “so you can focus on specific areas you want to do well in.” In general, how successful are test developers in providing such subscores in various testing programs? Not particularly (see Chapters 4 and 5), for such scores are typically based on few items and hence are not very reliable.

1.6 Subscore Specificity

Earlier we proposed that a “subscore is meaningful when it is reliable enough for its prospective use and contains information that is not adequately focused (or is overly diluted) in the total test score” (Haberman, Reference Haberman2008a). Having just discussed reliability, let us turn our attention to specificity.

If a subscore is merely reproducing the same information that is in the total score, we gain nothing from its use – indeed we are fooling ourselves into thinking that we have new, potentially valuable, information when we do not.Footnote ⁷ Continuing with our examination of the ASVAB as an example of a professionally polished test with subscores that potentially impact the lives and careers of its examinees, let us ask the extent to which its various composite scores provide information that is at least somewhat orthogonal to the information obtained from the total score.

The short answer is that they do not. In a sequence of reports in the early 1990s, Malcolm Ree, who for a decade was chairman of the ASVAB Technical Committee, repeatedly showed that the ASVAB and all its composite scores were essentially unidimensional, so any index formed by combining the ASVAB subtests yielded the same result (within the bounds of stochastic variation).

So, while the ASVAB does an admiral job of rank ordering examines and so improves the efficacy of selection for military training, the use of the myriad of indices based on their scores on various subtests is a chimera.

1.7 Is the ASVAB Like the Census?

The ASVAB is a serious test taken annually by more than a million people. When it was a linearly structured, paper-and-pencil test, it took more than three hours to complete. In the 1980s, the military undertook an ambitious project to transform the ASVAB to computerized administration and then to make it adaptive (W. A. Sands and his colleagues at the Navy Personnel Research and Development Center provide a full description in their 1997 technical report). This transformation allowed the test to be administered in only a little more than half the time with the same overall accuracy. The annual budget for the ASVAB is somewhere north of $20 million. The ASVAB is not the same size project as the census, but it is a serious test with serious goals. And although its developers desired efficacious subscores, it has been only partially successful in supplying them – they are reliable enough for many purposes but are not independent enough of one another to give guidance that is not more reliably given by the total score (AFQT).

Next, we shall compare the success of other testing programs with parallel goals.

1.8 Other Testing Programs That Want to Use Subscores

Over the past 20 years, due primarily to the passage of No Child Left Behind (and its sequel Every Student Succeeds), there has been a marked acceleration in K-12 testing in the United States. The act requires assessment in grades 3 through 8 with separate scores in mathematics, reading or language arts, and science. In a very real sense, the score on each of these topics is a subscore. They are calculated on various subpopulations of examinees, designated by race, geographic location, and so on, but are not reported back as individual scores to examinees and their families. Because what are calculated are mean scores over the subpopulations of interest, we probably need not be concerned about the reliability of the reported mean scores, at least for large subpopulations; however, concerns remain about the orthogonality of these subscores (see Chapter 4 for details). The stated goal of these subscores is to guide remediation of schools and districts. The aim is for each school to make at least adequate yearly progress (AYP) as defined by the act. Such goals are very serious indeed. Schools have at their disposal several, possibly draconian, actions to remediate less than AYP. These include “restructuring the school by: (1) reopening as a public charter school or (2) replacing all or most of the staff (which may include the principal) relevant to the AYP failure.”

There is no doubt that the goal is to have subscores that are accurate enough to make judgements about both the overall level of students’ performance in each of several subject matter areas and are also stable enough to estimate changes in them from one year to the next. This simply cannot be done effectively without reliable and orthogonal scores. The annual budget for this work is $250 million to cover roughly 50 million students, or $5 per student.

Compare this with the $40 per person cost of census or even the $20 per examinee cost of the ASVAB. We see that despite the seriousness of the consequences of poorly estimated subscores, the government has not seen fit to provide resources on the same level for them as they have for the census or for military testing.

1.9 Summing Up

1. I What’s Needed

   1. 1) Tests have been around for a long time (at least 4,000 years) and have been used to generate scores to provide evidentiary support for claims that users would like to make (e.g., this person should be admitted, be hired, be allowed to practice their profession).

   2. 2) But to provide valid support, a score must possess a number of critical characteristics. Two of these are:

      1. (i) Be relevant for its potential use

      2. (ii) Be reliable, remembering always that score reliability is immutably connected to the length of the test that generated it

   3. 3) For as long as there have been tests and test scores, there have been subtests and subscores.

   4. 4) For a subscore to be reliable enough to be useful, the test containing the subtest that generates that subscore must be longer, often much longer, and the more subscores, the greater the length of the test that contains it.

   5. 5) The extent to which a subscore yields unique information is directly related to the extent to which that subscore is orthogonal to the total test score.

2. II What We’ve Got

   1. 6) The inevitable conclusion is that constructing and administering a test whose score is valid and reliable is a straightforward task and has been accomplished successfully for millennia; but constructing such a test containing a number of subtests with scores that are also valid and reliable is much more difficult and expensive (expensive in both money and examinee time), and successful attempts are very rare indeed.

   2. 7) A model for success is the US decennial census, with small area estimates that are an analog to subscores. But obtaining such detailed information increases the cost of the census dramatically (to $42 per person counted).

   3. 8) A model for marginal success is the US military’s ASVAB program that, after almost a century of experience, evolved a combination of careful test construction and the optimization of item presentation that is possible with adaptive technology. It has thus been able to yield a set of eight subtests with associated subscores that are all reliable enough for most prospective uses. Unfortunately, it is not clear that any of the various composite scores calculated from these subscores (used for placement) yield any information that is distinct from any other composite, or indeed distinct from that obtained from the total score (which is, of course, considerably more reliable). At about $20 per person, the per examinee cost of the ASVAB is less than half that of census.

   4. 9) Another testing program we described briefly, which is typical of other, smaller, programs that also require subscores for various kinds of decisions, was one instigated by the 2001 No Child Left Behind Act (and its 2015 sequel Every Child Succeeds Act). There are potentially profound consequences for schools whose students’ subscores are unacceptably low (e.g., close the school or replace the principal and the faculty). Because of their known unreliability, these subscores are, sensibly, not reported to students or parents. But they are likely highly dependent on one another, so considering shortcomings on multiple subscores as replicated independent evidence is likely a mistake. The per student cost of these programs at $5 per person is about one fourth the ASVAB cost and one eighth that of census.

3. III What’s Next

• In the coming chapters, we will expand and deepen this introductory discussion of subscores.

• This will include, in Chapter 2, a description of how subscores are reported (and in Chapter 6 how we believe they should be reported).

• There is next (in Chapter 3) a careful description of how to tell if a subscore is worth reporting, including statistical measures of the potential value of a subscore.

• Then (in Chapter 4) we take the next step and discuss some viable subscores by giving examples of tests whose developers have actually succeeded in providing subscores that add value and the lessons that we can infer from these special tests. We then (in Chapter 5) provide some advice on what might be done when the subscores obtained ought not be reported. Finally, we move on to some potential avenues that research has shown can yield valid subscores.

• So far, though, nothing that we have learned allows one to construct a test with more information for less money/effort. Paraphrasing architect Philip Johnson, all good tests share one characteristic – they cost a lot of money or use a lot of examinee time, or usually, both.

• Thus, so far at least, there is still no free lunch.