Confidence Intervals for Reliability and Other Measures
01 / 2006 - unknown
Consumer traits such as attitude, behavior and intentions are frequently modelledas latent variables in marketing research, and are measured indirectly via observed indicators . An example is the annually held questionnaire Bank of the year by The Financial newspaper . Respondents have to answer questions on varioustopics such as products and services, granting of credits, quality of services, theirrelation manager and tariffs. These items measures respondent s attitude towardstheir bank. To examine the reliability of the questions for measuring each of these items, one usually estimates Cronbach s coefficient alpha. However, just reportingsuch a single quantity provides very little information. A crucial issue concernsthe evaluation of the value of that alpha, in other words: which values are high andwhich are low. In much marketing research rules of thumb are used, like morethan 0.8 is good. This practice is of course not very rigorous. Particularly, nothingis known about the accuracy of the measure. This problem can be overcomeby providing confidence intervals. The additional information included in the intervalallows for a more critical assessment of the measure. In this research projectwe stress the importance of confidence intervals accompanying single measuresthat function as summarizing quantity of the parameters of a model.Though it might be the most popular measure of scale reliability, the interpretationof Cronbach s coefficient alpha was rather arbitrarily till recently (see for instanceCortina (1993)). Adequacy of an obtained alpha was examined by means of Nunnally srecommendations (Nunnally and Bernstein, 1994). As a result of van Zylet al. (2000) coverage of statistical theory needed to construct confidence intervalsfor alpha, new emphasis is given on the superiority of inferential statistics overpoint estimators as coefficient alpha (Iacobucci and Duhachek (2003), Koning andFranses (2003), Duhachek and Iacobucci (2004), Duhachek et al. (2005)). Perhapsthis revived interest is due to the fact that van Zyl et al. (2000) derived the statisticaldistribution of alpha without further assumptions, while decades ago Kristof(1963) and Feldt (1965) derived this distribution under the very restrictive assumptionsof parallel measurements. Though it sounds very plausible when no furtherassumptions on the covariance matrix of the items are necessary, we believe thatignoring the structure in the covariance matrix will neglect valuable information.It should be stressed that between the extremely restrictive parallel model on theone hand, and the extremely permissive saturated model on the other hand, othermodels exist, see Lord and Novick (1968). As both extreme cases have their problems,this research project tries to bridge the gap between the parallel model andthe so-called saturated model. This research project aims at constructing confidenceintervals for alpha (and other reliability measures) in less restrictive models thanthe parallel model, namely tau-equivalent and congeneric model.In addition, this research project also aims to improve the reliability of tests bymeans of weights to yield a measure which is referred to as the maximal reliability(Li et al., 1996). Cronbach s coefficient alpha should be regarded as a lower boundof the reliability of a test, and in certain special situations it coincides with reliability,see Novick and Lewis (1967). In the parallel model Cronbach s alpha coincidewith reliability and also with maximal reliability. This holds no longer anymore forless restrictive models. It is aimed to derive the asymptotic distribution of maximalreliability under more plausible model assumptions to compute confidence intervalsas the importance of such inferential tests are already mentioned before.We also take special interest in models with correlated errors, such as panel designmodels and Multi-Trait Multi-Method (MTMM) models, (see Bagozzi et al.,1998). These models are special, as estimating alpha is no longer appropriate,2(see Green and Herschberger, 2000). We also take special interest in the relationof Cronbach s alpha in correspondence analysis shown in Lord (1958). It seemsnatural to extend the derived techniques in classical test theory into this field.