Alan L. Gross

Reading Rescue: An Effective Tutoring Intervention Model for Language-Minority Students Who Are Struggling Readers in First Grade

The Reading Rescue tutoring intervention model was investigated with 64 low–socioeconomic status, language-minority first graders with reading difficulties. School staff provided tutoring in phonological awareness, systematic phonics, vocabulary, fluency, and reading comprehension. Tutored students made significantly greater gains reading words and comprehending text than controls, who received a small-group intervention (d = 0.70) or neither intervention (d = 0.74). The majority of tutored students reached average reading levels whereas the majority of controls did not. Paraprofessionals tutored students as effectively as reading specialists except in skills benefiting nonword decoding. Paraprofessionals required more sessions to achieve equivalent gains. Contrary to conventional wisdom, results suggest that students make greater gains when they read text at an independent level than at an instructional level.

Restriction of range corrections when both distribution and selection assumptions are violated.

In validating a selection test (x) as a predictor of y, an incomplete xy data set must often be dealt with. A well-known correction formula is available for esti mating the xy correlation in some total group using the xy data of the selected cases and x data of the unse lected cases. The formula yields the ryx correlation (1) when the regression of y on x is linear and homosce dastic and (2) when selection can be assumed to be based on x alone. Although previous research has con sidered the accuracy of the correction formula when either Condition 1 or 2 is violated, no studies have considered the most realistic case where both Condi tions 1 and 2 are simultaneously violated. In the pres ent study six real data sets and five simulated selection models were used to investigate the accuracy of the correction formula when neither assumption is satis fied. Each of the data sets violated the linearity and/or homogeneity assumptions. Further, the selection models represent cases where selection is not a func tion of x alone. The results support two basic conclu sions. First, the correction formula is not robust to vi olations in Conditions 1 and 2. Reasonably small errors occur only for very modest degrees of selection. Secondly, although biased, the correction formula can be less biased than the uncorrected correlation for cer tain distribution forms. However, for other distribution forms, the corrected correlation can be less accurate than the uncorrected correlation. A description of this latter type of distribution form is given.

Relaxing the Assumptions Underlying Corrections for Restriction of Range.

It is generally believed that the correction for restriction of range will yield exact correlational values only when the regression of z on x is both linear and homoscedastic. However it is shown that the correction formula can hold even for non-linear heteroscedastic relationships. A simple sufficient condition for the validity of the formula is described. Further, simple conditions for predicting when the formula will overestimate and underestimate are also described. The results are demonstrated in terms of a numerical example.

Prediction In Future Samples Studied In Terms Of The Gain From Selection.

The gain from selection (GS) is defined as the standardized average performance of a group of subjects selected in a future sample using a regression equation derived on an earlier sample. Expressions for the expected value, density, and distribution function (DF) of GS are derived and studied in terms of sample size, number of predictors, and the prior distribution assigned to the population multiple correlation. The DF of GS is further used to determine how large sample sizes must be so that with probability .90 (.95), the expected GS will be within 90 percent of its maximum possible value. An approximately unbiased estimator of the expected GS is also derived.

Not Correcting for Restriction of Range can be Advantageous

In validating a selection test (x), complete test-criterion (y) scores are typically not available for all cases. Given this incomplete xy data set, one can estimate the population correlation using either the uncorrected correlation (computed using the data only from the selected group) or the so-called corrected correlation. Although the uncorrected value is always more biased than the corrected value, the former can have a substantially smaller expected mean square error when sample sizes are small, selection is extreme, and the population correlation is low.

A maximum likelihood approach to test validation with missing and censored dependent variables

A maximum likelihood approach is described for estimating the validity of a test (x) as a predictor of a criterion variable (y) when there are both missing and censoredy scores present in the data set. The missing data are due to selection on a latent variable (ys) which may be conditionally related toy givenx. Thus, the missing data may not be missing random. The censoring process in due to the presence of a floor or ceiling effect. The maximum likelihood estimates are constructed using the EM algorithm. The entire analysis is demonstrated in terms of hypothetical data sets.

The Differential Predictability of the College Performance of Males and Females

Multiple regression analyses using college freshman grade point average as the dependent variable and six high school scores as predictor variables were performed separately in 10 undergraduate colleges of the City University of New York. The multiple correlation coefficients, derived on the cross-validation samples, substantially reinforced the previously reported finding that females are more predictable than males in academic settings.

Estimating correlations with missing data, a bayesian approach

The posterior distribution of the bivariate correlation is analytically derived given a data set wherex is completely observed buty is missing at random for a portion of the sample. Interval estimates of the correlation are then constructed from the posterior distribution in terms of highest density regions (HDRs). Various choices for the form of the prior distribution are explored. For each of these priors, the resulting Bayesian HDRs are compared with each other and with intervals derived from maximum likelihood theory.

The Correction for Restriction of Range and Nonlinear Regressions: An Analytic Study

The effect of a nonlinear regression function on the accuracy of the restriction of range correction formula was investigated using analytic methods. Expressions were derived for the expected mean square error (EMSE) of both the correction formula and the squared correlation computed in the selected group, with re spect to their use as estimators of the population rela tionship. The relative accuracy of these two estimators was then studied as a function of the form of the regression, the form of the marginal distribution of x scores, the strength of the relationship, sample size, and the degree of selection. Although the relative ac curacy of the correction formula was comparable for both linear and concave regression forms, the correc tion formula performed poorly when the regression form was convex. Further, even when the regression is linear or concave, it may not be advantageous to employ the correction formula unless the xy relation ship is strong and sample size is large.

Interval Estimation of Bivariate Correlations With Missing Data on Both Variables: A Bayesian Approach:

The posterior distribution of the bivariate correlation (ρxy ) is analytically derived given a data set consisting N 1 cases measured on both x and y, N 2 cases measured only on x, and N 3 cases measured only ony. The posterior distribution is shown to be a function of the subsample sizes, the sample correlation (rxy ) computed from the N 1 complete cases, a set of four statistics which measure the extent to which the missing data are not missing completely at random, and the specified prior distribution for ρxy . A sampling study suggests that in small (N = 20) and moderate (N = 50) sized samples, posterior Bayesian interval estimates will dominate maximum likelihood based estimates in terms of coverage probability and expected interval widths when the prior distribution for ρxy is simply uniform on (0, 1). The advantage of the Bayesian method when more informative priors based on beta densities are employed is not as consistent.