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Chapter 1

Chapter 1 - Strauss.docx

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University of Toronto Scarborough
Zachariah Campbell

Chapter 1 Psychometrics in Neuropsychological Assessment The Normal Curve Basis of many commonly used statistical and psychometric models and is the assumed distribution for many psychological variables. Definition and Characteristics Unimodal, symmetrical, asymptotic at the tails. The ordinate (height of the curve at any point along the x-axis) is the proportion of persons within the sample who obtained a given score. Normal curve can also be referred to as a probability distribution. Relevance for Assessment As a frequency distribution, the area under any given segment of the normal curve indicates the frequency of observations or cases within that interval. • This provides psychologists with an estimate of the normality/abnormality of any given test score or range of scores o Normality – score falls in the center of the bell shape, where most of the scores are located o Abnormality – score falls at the ends of the bell shape, where there are few scores Z Scores and Percentiles Percentile: the percentage of scores that fall at or below a given test score Converting scores to percentiles – raw scores are ‘standardized’. Usually to Z scores z = (x – X)/SD x= measurement value (test score) X= the mean of the test score distribution SD= the standard deviation of the test score distribution Resulting distribution of Z scores has a mean of 0 and a SD of 1. Interpretation of Percentiles The relationship between raw or Z scores and percentiles is not linear. • A constant difference between raw or Z scores will be associated with a variable difference in percentile scores, as a function of the distance of the two scores from the mean. • This is due to the fact that there are proportionally more observations (scores) near the mean than there are farther from the mean o Otherwise, the distribution would be rectangular or non-normal Linear Transformations of Z Scores: T Scores and Other Standard Scores Linear transformation can be used to produce other standardized scores. T scores, Z scores, standard scores, and percentile equivalents are derived from samples. They are often treated as population values, any limitations of generalizability due to reference sample composition or testing circumstances must be taken into consideration when standardized scores are interpreted. The Meaning of Standardized Test Scores: Score Interpretation When comparing scores, it should be done when the distributions for tests that are being compared are approximately normal in the population. If standardized scores are to be compared, they should be derived from similar samples or (more ideally) from the same sample. Also when comparing scores, the reliability of the two measures must be considered and they intercorrelation before determining if a significance exists. • Relatively large disparities between standard scores may not actually reflect reliable differences and therefore may not be clinically meaningful. When test scores are not normally distributed, standardized scores may not accurately reflect actual population rank. Comparability across tests does not imply equality in meaning and relative importance of scores. Interpreting Extreme Scores In clinical practice, one may encounter standard scores that are either extremely low or high. The meaning/comparability of the scores depends on the characteristics of the normative sample from which they derive. Whenever extreme scores are being interpreted, examiners should verify that an examinee’s score falls within the range of raw scores in the normative sample. • If the normative sample size is substantially smaller than the estimated prevalence size and the examinee’s score falls outside the sample range, then considerable caution may be indicated in interpreting the percentile associated with the standardized score. When interpreting extreme scores, it depends on the properties of the normal samples involved. The Normal Curve and Test Construction A test with a normal distribution in the general population may show extreme skew or other divergence from normality when administered to a population that differs considerably from the average individual. Whether a test produces a normal distribution is also an important aspect of evaluating tests for bias across different populations. Depending on the characteristics of the construct being measured and the purpose for which a test is being designed, a normal distribution of scores may not be obtainable or desirable. For example: • The population distribution of the construct being measured may not be normally distributed • One may want only to identify and/or discriminate between persons at only one end of a continuum of abilities o The characteristics of only one side of the sample score distribution are critical while the characteristics on the other side of the distribution are not considered important o The measure may even be deliberately designed to have floor or ceiling effects Non-Normality It is not unusual for test score distributions to be markedly non-normal, even with large samples. The degree to which a given distribution approximates the underlying population distribution increases as the number of observations (N) increases and becomes less accurate as N decreases. • Larger sample will produce a more normal distribution only if the underlying population from which the sample is obtained is normal. o A large N does not correct for non-normality Small samples may yield non-normal distribution due to random sampling effects, even though the population from which the sample is drawn has a normal distribution. Factors that may lead to non-normal test score distributions: • Existence of discrete subpopulations within the general population with differing abilities • Ceiling or floor effects • Treatment effects that change the location of means, medians, and modes, and affect variability and distribution shape Skew Skew: formal measure of asymmetry in a frequency distribution that can be calculated using a specific formula • Third moment of a distribution (Mean is first movement, variance is the second movement) Normal distribution is perfectly symmetrical about the mean and has a skew of zero. Non-normal but symmetric distribution will have a skew value that is near zero • Negative skew: left tail of the distribution is heavier and often more elongated than the right tail • Positive skew: right tail of the distribution is heavier and often more elongated than the left tail When distributions are skewed, the mean and median are not identical because the mean will not be at the midpoint in rank. Z scores will not accurately translate into sample percentile rank values. This error in mapping increases as skew increases… Truncated Distributions Significant skew often indicates the presence of truncated distribution. • May occur when the range of score is restricted on one side but not the other Commonly seen with reaction time measures and on error scores. Floor/Ceiling Effects Def.: May be defined as the presence of truncated tails in the context of limitations in range of item difficulty. • High floor: large portion of the examinees obtain raw scores at or near the lowest possible score. o May indicate that the test lacks a sufficient number and range of easier items. • Low ceiling: high number of examinees obtain raw scores at or near the highest possible score These may significantly limit the usefulness of a measure. Multimodality and Other Types of Non-Normality Multimodality: presence of more than one ‘peak’ in a frequency distribution Uniform or near-uniform distribution: no or minimal peak and relatively equal frequency across scores. These distributions, may cause linearly transformed scores to be totally inaccurate with respect to actual sample/population percentile rank and should not be interpreted in that framework. Normalizing Test Scores When problematic score distributions occur, test developers employ ‘normalizing’ transformations in an attempt to correct departures from normality. • These do help but they don’t solve everything. They actually cause some more problems Test scores should only be normalized when: 1. They come from a large and representative sample 2. Any deviation from normality arises from defects in the test rather than characteristics of the sample Test makers should describe in detail the nature of any significant sample non-normality and the procedures used to correct it for derivation of standardized scores. Reasons for correction should also be justified. Extrapolation/Interpolation Norms can fall short in terms of range or cell size. In these cases, data are often extrapolated or interpolated using the existing score distribution. • Done using techniques like multiple regression • This often done with age extrapolations so that they go beyond the actual ages of the individuals in the sample. Measurement Precision: Reliability and Standard Error Psychological tests are not perfect and precise. Test scores are estimates of abilities or functions, with some degree of measurement error. • Each test differs in the precision of the scores that it produces A precise test can produce imprecise results if it’s administered: • In a nonstandard environment • In a nonoptimal environment • To an uncooperative examinee Definition of Reliability Def.: the consistency of measurement of a given test • Internal consistency reliability: consistency within a test • Test-retest reliability: consistency over time • Alternate form reliability: consistency across alternate forms of the test • Interrater reliability: consistency across different raters Reliability indicates the degree to which a test is free from measurement error. • Error actually consists of the multiple sources of variability that effect test scores Factors Affecting Reliability Reliability coefficien
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