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Reliability

From the perspective of classical test theory, an examinee's obtained test score (X) is composed of two components, a true score component (T) and an error component (E):

X=T+E

Reliability

The true score component reflects the examinee's status with regard to the attribute that is measured by the test, while the error component represents measurement error.

Measurement error is random error. It is due to factors that are irrelevant to what is being measured by the test and that have an unpredictable (unsystematic) effect on an examinee's test score.

Reliability

The score you obtain on a test is likely to be due both to the knowledge you have about the topics addressed by exam items (T) and the effects of random factors (E) such as the way test items are written, any alterations in anxiety, attention, or motivation you experience while taking the test, and the accuracy of your "educated guesses."

Reliability

Whenever we administer a test to examinees, we would like to know how much of their scores reflects "truth" and how much reflects error. It is a measure of reliability that provides us with an estimate of the proportion of variability in examinees' obtained scores that is due to true differences among examinees on the attribute(s) measured by the test.

Reliability

When a test is reliable, it provides dependable, consistent results and, for this reason, the term consistency is often given as a synonym for reliability (e.g., Anastasi, 1988).

Consistency = Reliability

The Reliability Coefficient

Ideally, a test's reliability would be calculated by dividing true score variance by the obtained (total) variance to derive a reliability index. This index would indicate the proportion of observed variability in test scores that reflects true score variability.

True Score Variance/Total Variance = Reliability Index

The Reliability Coefficient

A test's true score variance is not known, however, and reliability must be estimated rather than calculated directly.

There are several ways to estimate a test's reliability. Each involves assessing the consistency of an examinee's scores over time, across different content samples, or across different scorers.

The common assumption for each of these reliability techniques that consistent variability is true score variability, while variability that is inconsistent reflects random error.

The Reliability Coefficient

Most methods for estimating reliability produce a reliability coefficient, which is a correlation coefficient that ranges in value from 0.0 to + 1.0. When a test's reliability coefficient is 0.0, this means that all variability in obtained test scores is due to measurement error. Conversely, when a test's reliability coefficient is + 1.0, this indicates that all variability in scores reflects true score variability.

The Reliability Coefficient

The reliability coefficient is symbolized with the letter "r" and a subscript that contains two of the same letters or numbers (e.g., ''rxx'').

The subscript indicates that the correlation coefficient was calculated by correlating a test with itself rather than with some other measure.

The Reliability Coefficient

Regardless of the method used to calculate a reliability coefficient, the coefficient is interpreted directly as the proportion of variability in obtained test scores that reflects true score variability. For example, as depicted in Figure 1, a reliability coefficient of .84 indicates that 84% of variability in scores is due to true score differences among examinees, while the remaining 16% (1.00 - .84) is due to measurement error.

Figure 1. Proportion of variability in test scores

True Score Variability (84%)

Error (16%)

The Reliability Coefficient

Note that a reliability coefficient does not provide any information about what is actually being measured by a test!

A reliability coefficient only indicates whether the attribute measured by the test— whatever it is—is being assessed in a consistent, precise way.

Whether the test is actually assessing what it was designed to measure is addressed by an analysis of the test's validity.

The Reliability Coefficient

Study Tip: Remember that, in contrast to other correlation coefficients, the reliability coefficient is never squared to interpret it but is interpreted directly as a measure of true score variability. A reliability coefficient of .89 means that 89% of variability in obtained scores is true score variability.

Methods for Estimating Reliability

The selection of a method for estimating reliability depends on the nature of the test.

Each method not only entails different procedures but is also affected by different sources of error. For many tests, more than one method should be used.

1. Test-Retest Reliability:

The test-retest method for estimating reliability involves administering the same test to the same group of examinees on two different occasions and then correlating the two sets of scores. When using this method, the reliability coefficient indicates the degree of stability (consistency) of examinees' scores over time and is also known as the coefficient of stability.

Test-Retest Reliability

The primary sources of measurement error for test-retest reliability are any random factors related to the time that passes between the two administrations of the test.

These time sampling factors include random fluctuations in examinees over time (e.g., changes in anxiety or motivation) and random variations in the testing situation.

Memory and practice also contribute to error when they have random carryover effects; i.e., when they affect many or all examinees but not in the same way.

Test-Retest Reliability

Test-retest reliability is appropriate for determining the reliability of tests designed to measure attributes that are relatively stable over time and that are not affected by repeated measurement.

It would be appropriate for a test of aptitude, which is a stable characteristic, but not for a test of mood, since mood fluctuates over time, or a test of creativity, which might be affected by previous exposure to test items.

2. Alternate (Equivalent, Parallel) Forms Reliability:

To assess a test's alternate forms reliability, two equivalent forms of the test are administered to the same group of examinees and the two sets of scores are correlated.

Alternate forms reliability indicates the consistency of responding to different item samples (the two test forms) and, when the forms are administered at different times, the consistency of responding over time.

Alternate (Equivalent, Parallel) Forms Reliability

- The alternate forms reliability coefficient is also called the coefficient of equivalence when the two forms are administered at about the same time;
- and the coefficient of equivalence and stability when a relatively long period of time separates administration of the two forms.

Alternate (Equivalent, Parallel) Forms Reliability

The primary source of measurement error for alternate forms reliability is content sampling, or error introduced by an interaction between different examinees' knowledge and the different content assessed by the items included in the two forms (eg: Form A and Form B)

Alternate (Equivalent, Parallel) Forms Reliability

The items in Form A might be a better match of one examinee's knowledge than items in Form B, while the opposite is true for another examinee.

In this situation, the two scores obtained by each examinee will differ, which will lower the alternate forms reliability coefficient.

When administration of the two forms is separated by a period of time, time sampling factors also contribute to error.

Alternate (Equivalent, Parallel) Forms Reliability

Like test-retest reliability, alternate forms reliability is not appropriate when the attribute measured by the test is likely to fluctuate over time (and the forms will be administered at different times) or when scores are likely to be affected by repeated measurement.

Alternate (Equivalent, Parallel) Forms Reliability

- If the same strategies required to solve problems on Form A are used to solve problems on Form B, even if the problems on the two forms are not identical, there are likely to be practice effects.
- When these effects differ for different examinees (i.e., are random), practice will serve as a source of measurement error.
- Although alternate forms reliability is considered by some experts to be the most rigorous (and best) method for estimating reliability, it is not often assessed due to the difficulty in developing forms that are truly equivalent.

3. Internal Consistency Reliability:

Reliability can also be estimated by measuring the internal consistency of a test.

Split-half reliability and coefficient alpha are two methods for evaluating internal consistency. Both involve administering the test once to a single group of examinees, and both yield a reliability coefficient that is also known as the coefficient of internal consistency.

Internal Consistency Reliability

To determine a test's split-half reliability, the test is split into equal halves so that each examinee has two scores (one for each half of the test).

Scores on the two halves are then correlated. Tests can be split in several ways, but probably the most common way is to divide the test on the basis of odd- versus even-numbered items.

Internal Consistency Reliability

A problem with the split-half method is that it produces a reliability coefficient that is based on test scores that were derived from one-half of the entire length of the test.

If a test contains 30 items, each score is based on 15 items. Because reliability tends to decrease as the length of a test decreases, the split-half reliability coefficient usually underestimates a test's true reliability.

For this reason, the split-half reliability coefficient is ordinarily corrected using the Spearman-Brown prophecy formula, which provides an estimate of what the reliability coefficient would have been had it been based on the full length of the test.

Internal Consistency Reliability

Cronbach's coefficient alpha also involves administering the test once to a single group of examinees. However, rather than splitting the test in half, a special formula is used to determine the average degree of inter-item consistency.

One way to interpret coefficient alpha is as the average reliability that would be obtained from all possible splits of the test. Coefficient alpha tends to be conservative and can be considered the lower boundary of a test's reliability (Novick and Lewis, 1967).

When test items are scored dichotomously (right or wrong), a variation of coefficient alpha known as the Kuder-Richardson Formula 20 (KR-20) can be used.

Internal Consistency Reliability

Content sampling is a source of error for both split-half reliability and coefficient alpha.

- For split-half reliability, content sampling refers to the error resulting from differences between the content of the two halves of the test (i.e., the items included in one half may better fit the knowledge of some examinees than items in the other half);
- for coefficient alpha, content (item) sampling refers to differences between individual test items rather than between test halves.

Internal Consistency Reliability

Coefficient alpha also has as a source of error, the heterogeneity of the content domain.

A test is heterogeneous with regard to content domain when its items measure several different domains of knowledge or behavior.

Internal Consistency Reliability

The greater the heterogeneity of the content domain, the lower the inter-item correlations and the lower the magnitude of coefficient alpha.

Coefficient alpha could be expected to be smaller for a 200-item test that contains items assessing knowledge of test construction, statistics, ethics, epidemiology, environmental health, social and behavioral sciences, rehabilitation counseling, etc. than for a 200-item test that contains questions on test construction only.

Internal Consistency Reliability

The methods for assessing internal consistency reliability are useful when a test is designed to measure a single characteristic, when the characteristic measured by the test fluctuates over time, or when scores are likely to be affected by repeated exposure to the test.

They are not appropriate for assessing the reliability of speed tests because, for these tests, they tend to produce spuriously high coefficients. (For speed tests, alternate forms reliability is usually the best choice.)

4. Inter-Rater (Inter-Scorer, Inter-Observer) Reliability:

Inter-rater reliability is of concern whenever test scores depend on a rater's judgment.

A test constructor would want to make sure that an essay test, a behavioral observation scale, or a projective personality test have adequate inter-rater reliability. This type of reliability is assessed either by calculating a correlation coefficient (e.g., a kappa coefficient or coefficient of concordance) or by determining the percent agreement between two or more raters.

Inter-Rater (Inter-Scorer, Inter-Observer) Reliability

Although the latter technique is frequently used, it can lead to erroneous conclusions since it does not take into account the level of agreement that would have occurred by chance alone.

This is a particular problem for behavioral observation scales that require raters to record the frequency of a specific behavior.

In this situation, the degree of chance agreement is high whenever the behavior has a high rate of occurrence, and percent agreement will provide an inflated estimate of the measure's reliability.

Inter-Rater (Inter-Scorer, Inter-Observer) Reliability

Sources of error for inter-rater reliability include factors related to the raters such as lack of motivation and rater biases and characteristics of the measuring device.

An inter-rater reliability coefficient is likely to be low, for instance, when rating categories are not exhaustive (i.e., don't include all possible responses or behaviors) and/or are not mutually exclusive.

Inter-Rater (Inter-Scorer, Inter-Observer) Reliability

The inter-rater reliability of a behavioral rating scale can also be affected by consensual observer drift, which occurs when two (or more) observers working together influence each other's ratings so that they both assign ratings in a similarly idiosyncratic way.

(Observer drift can also affect a single observer's ratings when he or she assigns ratings in a consistently deviant way.) Unlike other sources of error, consensual observer drift tends to artificially inflate inter-rater reliability.

Inter-Rater (Inter-Scorer, Inter-Observer) Reliability

The reliability (and validity) of ratings can be improved in several ways:

- Consensual observer drift can be eliminated by having raters work independently or by alternating raters.
- Rating accuracy is also improved when raters are told that their ratings will be checked.
- Overall, the best way to improve both inter- and intra-rater accuracy is to provide raters with training that emphasizes the distinction between observation and interpretation (Aiken, 1985).

RELIABILITY AND VALIDITY

Study Tip: Remember the Spearman-Brown formula is related to split-half reliability and KR-20 is related to the coefficient alpha. Also know that alternate forms reliability is the most thorough method for estimating reliability and that internal consistency reliability is not appropriate for speed tests.

Factors That Affect The Reliability Coefficient

The magnitude of the reliability coefficient is affected not only by the sources of error discussed earlier, but also by the length of the test, the range of the test scores, and the probability that the correct response to items can be selected by guessing.

- Test Length
- Range of Test Scores
- Guessing

1. Test Length:

The larger the sample of the attribute being measured by a test, the less the relative effects of measurement error and the more likely the sample will provide dependable, consistent information.

Consequently, a general rule is that the longer the test, the larger the test's reliability coefficient.

Test Length

The Spearman-Brown prophecy formula is most associated with split-half reliability but can actually be used whenever a test developer wants to estimate the effects of lengthening or shortening a test on its reliability coefficient.

For instance, if a 100-item test has a reliability coefficient of .84, the Spearman-Brown formula could be used to estimate the effects of increasing the number of items to 150 or reducing the number to 50.

A problem with the Spearman-Brown formula is that it does not always yield an accurate estimate of reliability: In general, it tends to overestimate a test's true reliability (Gay, 1992).

Test Length

This is most likely to be the case when the added items do not measure the same content domain as the original items and/or are more susceptible to the effects of measurement error.

Note that, when used to correct the split-half reliability coefficient, the situation is more complex, and this generalization does not always apply: When the two halves are not equivalent in terms of their means and standard deviations, the Spearman-Brown formula may either over- or underestimate the test's actual reliability.

2. Range of Test Scores:

Since the reliability coefficient is a correlation coefficient, it is maximized when the range of scores is unrestricted.

The range is directly affected by the degree of similarity of examinees with regard to the attribute measured by the test.

Range of Test Scores

When examinees are heterogeneous, the range of scores is maximized.

The range is also affected by the difficulty level of the test items.

When all items are either very difficult or very easy, all examinees will obtain either low or high scores, resulting in a restricted range.

Therefore, the best strategy is to choose items so that the average difficulty level is in the mid-range (r = .50).

3. Guessing:

A test's reliability coefficient is also affected by the probability that examinees can guess the correct answers to test items.

As the probability of correctly guessing answers increases, the reliability coefficient decreases.

All other things being equal, a true/false test will have a lower reliability coefficient than a four-alternative multiple-choice test which, in turn, will have a lower reliability coefficient than a free recall test.

The Interpretation of Reliability

The interpretation of a test's reliability entails considering its effects on the scores achieved by a group of examinees as well as the score obtained by a single examinee.

Interpretation of Reliability Coefficient

The Reliability Coefficient: As discussed previously, a reliability coefficient is interpreted directly as the proportion of variability in a set of test scores that is attributable to true score variability.

A reliability coefficient of .84 indicates that 84% of variability in test scores is due to true score differences among examinees, while the remaining 16% is due to measurement error.

While different types of tests can be expected to have different levels of reliability, for most tests in the social sciences, reliability coefficients of .80 or larger are considered acceptable.

The Interpretation of Reliability

When interpreting a reliability coefficient, it is important to keep in mind that there is no single index of reliability for a given test.

Instead, a test's reliability coefficient can vary from situation to situation and sample to sample. Ability tests, for example, typically have different reliability coefficients for groups of individuals of different ages or ability levels.

Interpretation of Standard Error of Measurement

While the reliability coefficient is useful for estimating the proportion of true score variability in a set of test scores, it is not particularly helpful for interpreting an individual examinee's obtained test score.

When an examinee receives a score of 80 on a 100-item test that has a reliability coefficient of .84, for instance, we can only conclude that, since the test is not perfectly reliable, the examinee's obtained score might or might not be his or her true score.

Interpretation of Standard Error of Measurement

A common practice when interpreting an examinee’s obtained score is to construct a confidence interval around that score.

The confidence interval helps a test user estimate the range within which an examinee's true score is likely to fall given his or her obtained score.

This range is calculated using the standard error of measurement, which is an index of the amount of error that can be expected in obtained scores due to the unreliability of the test. (When raw scores have been converted to percentile ranks, the confidence interval is referred to as a percentile band.)

Interpretation of Standard Error of Measurement

The following formula is used to estimate the standard error of measurement:

Formula 1: Standard Error of Measurement

SEmeas = SDx *(1 – rxx)1/2

Where:

SEmeas = standard error of measurement

SDx = standard deviation of test scores

rxx= reliability coefficient

Interpretation of Standard Error of Measurement

As shown by the formula, the magnitude of the standard error is affected by two factors:

- the standard deviation of the test scores (SDx), and
- the test's reliability coefficient (rxx).
The lower the test's standard deviation and the higher its reliability coefficient, the smaller the standard error of measurement (and vice versa).

Interpretation of Standard Error of Measurement

Because the standard error is a type of standard deviation, it can be interpreted in terms of the areas under the normal curve.

With regard to confidence intervals, this means that a 68% confidence interval is constructed by adding and subtracting one standard error to an examinee's obtained score; a 95% confidence interval is constructed by adding and subtracting two standard errors; and a 99% confidence interval is constructed by adding and subtracting three standard errors.

Interpretation of Standard Error of Measurement

Example: A psychologist administers a interpersonal assertiveness test to a sales applicant who receives a score of 80. Since the test's reliability is less than 1.0, the psychologist knows that this score might be an imprecise estimate of the applicant's true score and decides to use the standard error of measurement to construct a 95% confidence interval. Assuming that the test’s reliability coefficient is .84 and its standard deviation is 10, the standard error of measurement is equal to 4.0:

SEmeas = SDx*(1 – rxx)1/2 = 10 (1 - .84)1/2 = 10(.4) = 4.0

The psychologist constructs the 95% confidence interval by adding and subtracting two standard errors from the applicant's obtained score: 80 + 2(4.0) = 72 to 88. This means that there is a 95% chance that the applicant's true score falls somewhere between 72 and 88.

Interpretation of Standard Error of Measurement

One problem with the standard error is that measurement error is not usually equally distributed throughout the range of test scores.

Use of the same standard error to construct confidence intervals for all scores in a distribution can, therefore, be somewhat misleading.

To overcome this problem, some test manuals report different standard errors for different score intervals.

Estimating True Scores from Obtained Scores

As discussed earlier, because of the effects of measurement error, obtained test scores tend to be biased (inaccurate) estimates of true scores.

More specifically, scores above the mean of a distribution tend to overestimate true scores, while scores below the mean tend to underestimate true scores.

Estimating True Scores from Obtained Scores

Moreover, the farther from the mean an obtained score is, the greater this bias.

Rather than constructing a confidence interval, an alternative (but less used) method for interpreting an examinee's obtained test score is to estimate his/her true score using a formula that takes into account this bias by adjusting the obtained score using the mean of the distribution and the test's reliability coefficient.

Estimating True Scores from Obtained Scores

For example, if an examinee obtains a score of 80 on a test that has a mean of 70 and a reliability coefficient of .84, the formula predicts that the examinee's true score is 78.2.

T’=a + bX

=(1-rxx )X + rxx X

T’=(1-.84) x 70 + .84 x 80

=.16 x 70 + .84 x 80

=11.2 + 67=78.2

The Reliability of Difference Scores

A test user is sometimes interested in comparing the performance of an examinee on two different tests or subtests and, therefore, computes a difference score. An educational psychologist, for instance, might calculate the difference between a child's WISC-IV Verbal and Performance scores.

The Reliability of Difference Scores

When doing so, it is important to keep in mind that the reliability coefficient for the difference scores can be no larger than the average of the reliabilities of the two tests or subtests:

If Test A has a reliability coefficient of .95 and Test B has a reliability coefficient of .85, this means that difference scores calculated from the two tests will have a reliability coefficient of .90 or less.

The exact size of the reliability coefficient for difference scores depends on the degree of correlation between the two tests: The more highly correlated the tests, the smaller the reliability coefficient (and the larger the standard error of measurement).

Validity

Validity refers to a test's accuracy. A test is valid when it measures what it is intended to measure. The intended uses for most tests fall into one of three categories, and each category is associated with a different method for establishing validity:

- The test is used to obtain information about an examinee's familiarity with a particular content or behavior domain: content validity.
- The test is administered to determine the extent to which an examinee possesses a particular hypothetical trait: construct validity.
- The test is used to estimate or predict an examinee's standing or performance on an external criterion: criterion-related validity.

Validity

For some tests, it is necessary to demonstrate only one type of validity; for others, it is desirable to establish more than one type.

For example, if an arithmetic achievement test will be used to assess the classroom learning of 8th grade students, establishing the test's content validity would be sufficient. If the same test will be used to predict the performance of 8th grade students in an advanced high school math class, the test's content and criterion-related validity will both be of concern.

Note that, even when a test is found valid for a particular purpose, it might not be valid for that purpose for all people. It is quite possible for a test to be a valid measure of intelligence or a valid predictor of job performance for one group of people but not for another group.

Content Validity

A test has content validity to the extent that it adequately samples the content or behavior domain that it is designed to measure.

If test items are not a good sample, results of testing will be misleading.

Although content validation is sometimes used to establish the validity of personality, aptitude, and attitude tests, it is most associated with achievement-type tests that measure knowledge of one or more content domains and with tests designed to assess a well-defined behavior domain.

Adequate content validity would be important for a statistics test and for a work (job) sample test.

Content Validity

Content validity is usually "built into" a test as it is constructed through a systematic, logical, and qualitative process that involves clearly identifying the content or behavior domain to be sampled and then writing or selecting items that represent that domain.

Once a test has been developed, the establishment of content validity relies primarily on the judgment of subject matter experts.

If experts agree that test items are an adequate and representative sample of the target domain, then the test is said to have content validity.

Content Validity

Although content validation depends mainly on the judgment of experts, supplemental quantitative evidence can be obtained.

If a test has adequate content validity:

- a coefficient of internal consistency will be large;
- the test will correlate highly with other tests that purport to measure the same domain; and
- pre-/post-test evaluations of a program designed to increase familiarity with the domain will indicate appropriate changes.

Content Validity

Don’t confuse Content validity with Face validity.

Content validity refers to the systematic evaluation of a test by experts who determine whether or not test items adequately sample the relevant domain, while face validity refers simply to whether or not a test "looks like" it measures what it is intended to measure.

Although face validity is not an actual type of validity, it is a desirable feature for many tests. If a test lacks face validity, examinees may not be motivated to respond to items in an honest or accurate manner. A high degree of face validity does not, however, indicate that a test has content validity.

Construct Validity

When a test has been found to measure the hypothetical trait (construct) it is intended to measure, the test is said to have construct validity. A construct is an abstract characteristic that cannot be observed directly but must be inferred by observing its effects. intelligence, mechanical aptitude, self-esteem, and neuroticism are all constructs.

There is no single way to establish a test's construct validity. Instead, construct validation entails a systematic accumulation of evidence showing that the test actually measures the construct it was designed to measure. The various methods used to establish this type of validity each answer a slightly different question about the construct and include the following:

Construct Validity

- Assessing the test's internal consistency: Do scores on individual test items correlate highly with the total test score; i.e., are all of the test items measuring the same construct?
- Studying group differences: Do scores on the test accurately distinguish between people who are known to have different levels of the construct?
- Conducting research to test hypotheses about the construct: Do test scores change, following an experimental manipulation, in the direction predicted by the theory underlying the construct?

Construct Validity

- Assessing the test's convergent and discriminant validity: Does the test have high correlations with measures of the same trait (convergent validity) and low correlations with measures of unrelated traits (discriminant validity)?
- Assessing the test's factorial validity: Does the test have the factorial composition it would be expected to have; i.e., does it have factorial validity?

Construct Validity

Construct validity is said to be the most theory-laden of the methods of test validation.

The developer of a test designed to measure a construct begins with a theory about the nature of the construct, which then guides the test developer in selecting test items and in choosing the methods for establishing the test's validity.

Construct Validity

For example, if the developer of a creativity test believes that creativity is unrelated to general intelligence, that creativity is an innate characteristic that cannot be learned, and that creative people can be expected to generate more alternative solutions to certain types of problems than non-creative people, she would want to determine the correlation between scores on the creativity test and a measure of intelligence, to see if a course in creativity affects test scores, and find out if test scores distinguish between people who differ in the number of solutions they generate to relevant problems

Construct Validity

Note that some experts consider construct validity to be the most basic form of validity because the techniques involved in establshing construct validity overlap those used to determine if a test has content or criterion-related validity.

Indeed, Cronbach argues that "all validation is one, and in a sense all is construct validation."

Construct Validity

Convergent and Discriminant Validity:

As mentioned earlier one way to assess a test's construct validity is to correlate test scores with scores on measures that do and do not purport to assess the same trait.

High correlations with measures of the same trait provide evidence of the test's convergent validity, while low correlations with measures of unrelated characteristics provide evidence of the test's discriminant (divergent) validity.

Construct Validity

The multitrait-multimethod matrix (Campbell & Fiske, 1959) is used to systematically organize the data collected when assessing a test's convergent and discriminant validity.

The multitrait-multimethod matrix is a table of correlation coefficients, and, as its name suggests, it provides information about the degree of association between two or more traits that have each been assessed using two or more methods.

When the correlations between different methods measuring the same trait are larger than the correlations between the same and different methods measuring different traits, the matrix provides evidence of the test's convergent and discriminant validity.

Multitrait-multimethod matrix

Example: To assess the construct validity of the interpersonal assertiveness test, a psychologist administers four measures to a group of salespeople: ( 1 ) the test of interpersonal assertiveness; (2) a supervisor's rating of interpersonal assertiveness; (3) a test of aggressiveness; and (4) a supervisor's rating of aggressiveness.

The psychologist has the minimum data needed to construct a multitrait-multimethod matrix: She has measured two traits that she believes are unrelated (assertiveness and aggressiveness), and each trait has been measured by two different methods (a test and a supervisor-s rating). The psychologist calculates correlation coefficients for all possible pairs of scores on the four measures and constructs the following multitrait-multimethod matrix (the upper half of the table has not been filled in because it would simply duplicate the correlations in the lower half):

Assertiveness Test

B1

Aggressiveness

Test

A2

Assertiveness

Rating

B2

Aggressiveness

Rating

A1

rA1A1 (.93)

B1

rB1A1 (.13)

rB1B1 (.91)

A2

rA2A1 (.71)

rA2B1 (.09)

rA2A2 (.86)

B2

rB2A1 (.04)

rB2B1 (.68)

rB2A2 (.16)

rB2B2 (.89)

Multitrait-multimethod matrixMultitrait-multimethod matrix

All multitrait-multimethod matrices contain four types of correlation coefficients:

- Monotrait-monomethod coefficients ("same trait-same method")
- Monotrait-heteromethod coefficients ("same trait-different methods")
- Heterotrait-monomethod coefficients ("different traits-same method")
- Heterotrait-heteromethod coefficients ("different traits-different methods“)

Monotrait-monomethod coefficients

(or the "same trait-same method")

The monotrait-monomethod coefficients (coefficients in parentheses in the previous matrix) are reliability coefficients:

They indicate the correlation between a measure and itself.

Although these coeffcients are not directly relevant to a test's convergent and discriminant validity, they should be large in order for the matrix to provide useful information.

Monotrait-heteromethod coefficients

(or "same trait-different methods"):

These coefficients (coefficients in rectangles) indicate the correlation between different measures of the same trait.

When these coefficients are large, they provide evidence of convergent validity.

Heterotrait-monomethod coefficients

(or "different traits-same method"):

These coefficients (coefficients in ellipses) show the correlation between different traits that have been measured by the same method.

When the heterotrait-monomethod coefficients are small, this indicates that a test has discriminant validity.

Heterotrait-heteromethod coefficients

(or "different traits-different methods"):

The heterotrait-heteromethod coefficients (underlined coefficients) indicate the correlation between different traits that have been measured by different methods.

These coefficients also provide evidence of discriminant validity when they are small

Multitrait-multimethod matrix

Note that, in a multitrait-multimethod matrix, only those correlation coefficients that include the test that is being validated are actually of interest.

In our example matrix, the correlation between the rating of interpersonal assertiveness and the rating of aggressiveness (r = .16) is a heterotrait-monomethod coefficient, but it isn't of interest because it doesn't provide information about the interpersonal assertiveness test.

Multitrait-multimethod matrix

Also, the number of correlation coefficients that can provide evidence of convergent and discriminant validity depends on the number of measures included in the matrix.

In the example, only four measures were included (the minimum number), but there could certainly have been more.

Multitrait-multimethod matrix

Example: Three of the correlations in our multitrait-multimethod matrix are relevant to the construct validity of the interpersonal assertiveness test.

The correlation between the assertiveness test and the assertiveness rating (monotrait-heteromethod coefficient) is .71. Since this is a relatively high correlation, it suggests that the test has convergent validity.

Multitrait-multimethod matrix

The correlation between the assertiveness test and the aggressiveness test (heterotrait-monomethod coefficient) is .13 and the correlation between the assertiveness test and the aggressiveness rating (heterotrait-heteromethod coefficient) is .04.

Because these two correlations are low, they confirm that the assertiveness test has discriminant validity. This pattern of correlation coefficients confirms that the assertiveness test has construct validity.

Note that the monotrait-monomethod coefficient for the assertiveness test is .93, which indicates that the test also has adequate reliability. (The other correlations in the matrix are not relevant to the psychologist's validation study because they do not include the assertiveness test.)

Assertiveness Test

B1

Aggressiveness

Test

A2

Assertiveness

Rating

B2

Aggressiveness

Rating

A1

rA1A1 (.93)

B1

rB1A1 (.13)

rB1B1 (.91)

A2

rA2A1 (.71)

rA2B1 (.09)

rA2A2 (.86)

B2

rB2A1 (.04)

rB2B1 (.68)

rB2A2 (.16)

rB2B2 (.89)

Multitrait-multimethod matrixConstruct Validity

Factor Analysis: Factor analysis is used for several reasons including identifying the minimum number of common factors required to account for the intercorrelations among a set of tests or test items, evaluating a test’s internal consistency, and assessing a test’s construct validity.

When factor analysis is used in the latter purpose, a test is considered to have construct (factorial) validity when it correlates highly only with the factor(s) that it would be expected to correlate with.

Reliability & Validity

Which is plausible?

- An instrument that is reliable, but not valid?
OR

- An instrument that is valid, but not reliable?
Why is this the case?

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