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This means that the items possesses. Thus if Y is high, all the item scores will ent levels of the attribute it represents , scores on all the items will vary with it of a given error term only affects the score of one item. Thus, the error terms variable affects all of the items and thus they are correlated. The error terms, in All of the variation in items that is due to the latent variable, Y, is shared or So, the latent the same error source, under our classical measurement assumptions. The value. So, each item and, by implication the scale as of the variances of the individual items.

Thus the covariance matrix gives us The variance, a;, of the k-item scale equals the sum of all matrix elements. According to the defmition of reliability, alpha should matrix is, by definition, the variance ofY, the scale made up of the individual The entries along the main diagonal are the variances of the individual items two values can be given a conceptual interpretation. The sum of the whole that we refer to as error. It follows that the total variance for each item, and variances, r,a;, computed by summing entries along the main diagonal.

These items. I noted covariances of items with themselves. Each variance contains information about only one item. In other two items; words, that item, Figure 3. The oncommunal item's variance does shaded a1,3 area each represents information that is based on a single item, A variance-covariance matrix showing that the variances along the main diagonal shaded area are noncommunal, along not quantify regions within the triangular borders are, together, the communal portion. Thus the elements in the irrespective of what causes it. The off-diagonal elements of the the extent of shared variance, merely the amount of dispersion in the scores for variation shared among items.

Within that single item, some of its variation not joint covariance matrix and hence the total variance of Y consist of covariation diagonal is the noncommunal portion of the matrix, and the two off-diagonal cerning items considered individually.

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Figure 3. It thus follows that we can express the proportion of involved. Even if there were no need to calculate these variances for each individual item, i, were probably operations that had already been done to do calculations. Why not just compute the sum of the off-diagonal elements This ratio corresponds to the sum of the diagonal values in the covariance joint, or communal, variances elements.

Thus we can express the ratio ofnonjoint variation to total variation covariance matrix. Thus, a formula that quantifies scale, the choice would be between computing 21 variances one for each item subtraction from 1 is a legacy of the days when computers were not available communal item directly computing the sum of off-diagonal elements. To do this, we multiply by k!! To adjust our calculations so that the ratio expresses the relative mag- communal i. The number that are com- possible values for alpha to between 0. In the five- The number of elements in the matrix that are non- would at first blush seem to capture the definition of alpha, that is, the com- common source, which we presume reflects the true score of the latent vari- able.

We still need one more correction, however. This need becomes appar- To summarize, a measure's reliability equals the proportion of total vari- 35 or. Why is this so? This the total variance ance among its items that is due to the latent variable and thus is communal. Such an arrangement should result in perfect reliability. The correlation ent if we consider what would happen if we had, say, five perfectly correlated k. The fraction in our last formula nitudes rather than the numbers of terms that are summed in the numerator and denominator, we multiply the entire expression representing the proportion of Readers may want to do the mental arithmetic for matrices of other sizes.

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Consider the individual values. However, the sum of these individual values is identical to rather than covariances. Actually, it uses r, the average interitem correlation. Therefore, the numerator of the tenn tence; the average of 10 values that sum to 50 has to be 5, and 10 times 5 equals the tion is a covariances ized formula.

Consequently, we can replace c with the aver- about item means and variances in the computation process, because covari- does not retain the original scaling metric of the items. Recall that a correla- The two different formulas, one based on covariances and the other on cor- standardizing, an average of covariances is identical to an average of correla- metric and thus weighted equally in the computation of alpha by the standard- 37 formulas for alpha, respectively. The raw score formula preserves information relations, are sometimes referred to as the raw score and standardized score 1 from the numerator of the right Recall that the formula we are striving for involves correlations rather than items have markedly different variances, those with larger variances will be its important uses will be illustrated in the section of this chapter dealing with used to compute alpha.

The standardized score formula based on correlations common formula is desired. As we shall see in later chapters, recommended better depends on the specific context and whether and v with 1. This yields the correlation-based After Cross-canceling k from the numerator of the left term and denominator of This formula is known as the Spearman-Brown prophecy formula, and one of ances are based on values that retain the original scaling of the raw data.

When A at producing equivalent item variances are not followed, versus these precise standardized and raw alpha values differ appreciably e. On the other hand, when procedures aimed followed, there is typically little difference in the alpha coefficients computed I error, the effect size e. Researchers might do well to weigh the statistical power for a given sample size or allow a smaller sample size to statistical test can be increased by increasing the sample size.

The probability of detecting such a difference i. Substituting a highly yield equivalent power , relative to less reliable measures.

To have a specified measurement. In a either form with itself, as each alternate form is equivalent to the other. Under these conditions, that both forms of the scale have identical alphas, means, and variances and of items that has been divided more or less arbitrarily into two subsets that measure the same phenomenon. In essence, parallel forms consist of one set and However, there are other reliability estimates that apply the same sort of logic physicians, then administered both sets of items to a group of patients and, finally, correlated the scores from one set of items with the scores from the versions of a scale that conform strictly to the assumptions of parallel tests.

Lipsey provides a more comprehensive discussion of statistical forms. For example, assume that a researcher first developed two equivalent with same people complete both parallel If two strictly parallel forms of a scale exist, then the correlation between other set. This correlation would be the alternate forms reliability. Recall that forms do an equally good job of measuring the latent variable. This implies a take the set of items that makes up a single scale i. Because alternate forms are essentially made up of a of. For example, if the items making up the scale in because there are a variety of ways in which the scale can be split in half.

One split may be problematic, however, because factors othe1 method is to compare the first half of the items to the second half. This type lower the correlation between halves because of the order in which the scale would then differ systematically between the two halves and would thus make absence of the attribute in question. The two halves of the scale would then be ity of a questionnaire from front to back.

This ensures that each of the two subsets of items consists of an equal than the value of the latent variable in other words, sources of error mighl A reliability measure of this type is called a split-half reliability.

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Split-haU order common to achievement tests, for example , this method avoids many ol balanced halves and random halves. In the former case, one would identify scale. However, the dissimilarity would not be so much a items were presented and not because of the quality of the scale items. Two taken to avoid this. The reasoning behind both split-halves and alternate-forms Although when we initially discussed that model, we regarded each item as strength of the relationships each item to one of the two subsets that will eventually be correlated with one the number of items, the number of characteristics of concern, and the degree balance another.

This would be the case, for example, if there were more long latter characteristic items, varying along several interrelated dimensions, will yield comparable instance, the subset of odd-numbered items is compared to the even-numbered groupings through randomization is unrealistic. On the other hand, randomly it may be difficult to be impossible to balance the proportion of one without making it impossible tc between the tests.

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If the path values have to be equal and they do, under the and so on. However, when considering multiple item characteristics, it mighi variable to each. Thus the product of these paths' values equals the correlation The only route linking the two consists of the causal paths from the latent another to compute the reliability estimate.

Hoping that a small number of. If two scale. An estimate of the reliability of the versions of a scale in the alternate forms case, they are two half-scales in the of variance in either test that is influenced by the latent variable. This, in turn, path assuming that it is a standardized path coefficient is also the proportion tests square of the path value from latent variable to either test. The square of that chapter. Thus, the reliability esti- reliability as the proportion of variance attribut- may or may not have to do with the error-proneness of the measurement pro- desire for control when interacting with physicians, our hypothetical investi- despite variations in anxiety.

The stability in scores, reflected in a high corre- interest has changed.

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For example, if a purported anxiety measure was influ- should exert comparable influence on observed scores on two or more occa- make it apparent that, all else being equal, a longer scale will always be more 43 measure patients' You could then recompute the reliability for the whole item scale by using stability of a measure, or how constant scores remain from one occasion to trations of the scale. Consequently, the correlation of scores obtained across this.

A quick look at the Spearman-Brown formula should Another two-score method of computing reliability involves the temporal enced by social desirability as well as anxiety, scores might remain constant truly same individuals should represent the reflects some meaningful construct, it should assess that construct comparably The problem with this reasoning is that what happens to the scores over time The rationale occasion might gator developed only a single set.

Those items could be given to one group of cause them to yield temporally stable responses even when the construct of. Of these factors, fatigue effects that cause items to be misread , and d temporal instability due only the fourth is unreliability. These authors also note that, although methods This is not to say that demonstrating temporal stability is unimportant.