Cronbach’s Alpha assumptions:

1- That the indicators (items) are essentially tau-equivalent o This implies that each item is an equally strong indicator of the true score scores, but the ma differ by a constant

o Basically this implies that the items can have different means

2- That each item’s error term is uncorrelated with every other item’s error term

3- That the error scores are uncorrelated with the true scores o An assumption associated with all forms of reliability

4- The items used to generate the questionnaire must only measure one attribute or construct o If this is violated, Cronbach’s alpha cannot be used

Standardised coefficient alpha:

  • You apply it to scores that have been converted from a raw score to a standardised score
  • g. if you had z-scores and you wanted to calculate the level of internal consistency associated with a composite which consisted of a sum of two or more z-scores, you would use the standardized version of coefficient alpha

Types of reliability assumption models:

  • Parallel- equal true score variance, equal means, equal error variance
  • Tau-equivalent- equal true score variance, equal means, unequal error variances
  • Essentially tau-equivalent- equal true score variance, unequal means, unequal error variances
  • Congeneric- unequal true score variance (but all greater than zero), unequal means, unequal error variances

What Cronbach’s alpha is not?

  • Cronbach’s alpha represents the ratio of true score variance to total variance
  • It does not imply that a collection of items measure one psychological attribute (or dimension) and one psychological attribute only.
  • It is fully possible for a collection of items to measure more than one psychological attribute, but still be associated with a reasonably high Cronbach’s alpha
  • Argued that CA assumes that the collection of items measure only one attribute o This can be tested using factor analysis

Factors affecting reliability:

  • Will adding this item reduce the mean inter-item correlation?

Sample homogeneity:

  • A more homogenous sample will yield lower reliability estimates than a heterogeneous sample o This is true of all correlations, not just statistics like Cronbach’s alpha which is fundamentally based on correlations

 

Importance:

Mental retardation is defined as a person with an IQ score of 70 or less

  • Some US states, death penalty cannot be administered to mental retards

Point estimates and confidence intervals:

  • The standard error of measurement represents the amount of error ‘around’ a point estimate in standard deviation form
  • A point estimate is one’s best guess of what a person’s score is on the test o In the context of psychometrics, it’s the score you get from the test o However a confidence interval can be estimated around a point-estimate
  • A confidence interval reflects a range of values that is often interpreted as a range in which the true score is likely to fall
  • Sem = so √(1-Rxx) o So = standard deviation of the test o Rxx = reliability estimate
  • Typically, people calculate the 5% confidence interval around a point estimate
  • rnce the standard error of estimate has been calculated, you can use this info to calculate a confidence interval around the point estimate

What confidence level should be used?

  • You probably want to use a 99% CI for a lift or death penalty
  • You should keep in mind that although high levels of reliability are desired, they are not the only consideration
  • rften have to sacrifice some reliability for greater validity

Reliability standards:

  • Using the standardised coefficient alpha formula, you can work out how large the mean inter-item correlation must be to achieve a reliability of a desired confidence interval

krii ‘

  • Rxx=1+(k−1)rii’

Two types of correlations:

  • rbserved score correlation:
    • The correlation you get based on the data you have
    • Will be compromised to the degree to which there is measurement error in your data, when scores are associated with less than perfect reliability
    • In practice this means the maximum correlation between two sets is not 1.0 but often less than that

 

  • The maximum correlation between two variables- rmax=√RxxRyy o The correlation for attenuation formula is the ration of the observed correlation to

rxO yO

the square root of the product of the reliabilities- rxt yt=√RxxRyy

  • True score correlation:
    • A hypothetical correlation you can estimate, if you know the reliability associate with the scores
    • Not compromised by measurement error