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. Author manuscript; available in PMC: 2024 Jan 1.
Published in final edited form as: Sci Stud Read. 2022 Dec 22;27(1):67–81. doi: 10.1080/10888438.2022.2118057

A Bayesian Probabilistic Framework for Identification of Individuals with Dyslexia

Richard K Wagner 1, Jerad Moxley 2, Chris Schatschneider 1, Fotena A Zirps 1
PMCID: PMC9851422  NIHMSID: NIHMS1857233  PMID: 36685047

Abstract

Purpose:

Bayesian-based models for diagnosis are common in medicine but have not been incorporated into identification models for dyslexia. The purpose of the present study was to evaluate Bayesian identification models that included a broader set of predictors and that capitalized on recent developments in modeling the prevalence of dyslexia.

Method:

Model-based meta-analysis was used to create a composite correlation matrix that included common predictors of dyslexia such as decoding, phonological awareness, oral language, but also included response to intervention (RTI) and family risk for dyslexia. Bayesian logistic regression models were used to predict poor reading comprehension, unexpectedly poor reading comprehension, poor decoding, and unexpectedly poor decoding, all at two levels of severity.

Results:

Most predictors made independent and substantial contributions to prediction, supporting models of dyslexia that rely on multiple rather than single indicators. RTI was the strongest predictor of poor reading comprehension and unexpectedly poor reading comprehension. Phonological awareness was the strongest predictor of poor decoding and unexpectedly poor decoding, followed closely by family risk.

Conclusion:

Bayesian-based models are a promising tool for implementing multiple-indicator models of identification. Ideas for improving prediction and implications for theory and practice are discussed.

Operational definitions of dyslexia that are based primarily on a single criterion have insufficient reliability for use in research or in practice (Wagner et al., in press). This is true regardless of whether the operational definition features response to instruction and intervention, IQ-achievement discrepancy, or low achievement (Barth et al, 2008; Fletcher et al., 2019; D. Fuchs et al., 1997; Pennington et al., 2019; Schatschneider et al., 2016; Waesche et al., 2011).

The unreliability emerges when diagnosis is framed as a dichotomous yes or no decision, which requires placing a cut-point on an underlying continuous distribution, and because of measurement error in any observed criterion (Francis et al., 2005). For relatively low base-rate conditions such as developmental dyslexia, many affected individuals will fall near the cut-off. Measurement error will result in some affected individuals scoring on the other side of the cut-off on subsequent measurement occasions. Because unaffected individuals can score far away from the cut-off, measurement error is less likely to change their classification. And because percentage agreement and kappa are averages of agreement for both decisions, they overestimate the accuracy of the decision that dyslexia is present (Waesche et al., 2011; Wagner et al, 2020).

Two potential solutions to the problem of unreliability in identification of individuals with dyslexia are the focus of the current study. The first is moving from models based primarily on single criteria to models based on multiple criteria, and the second is to move from dichotomous decisions to probabilistic decision-making.

Improving Identification Accuracy Using Multiple Criteria

One way to reduce measurement error is to have multiple criteria rather than to rely primarily on a single criterion for identification (Francis et al., 2005; Macmann et al., 1989). Consider some examples of proposed multiple-indicators models.

A hybrid model was proposed that consisted of three main criteria: low achievement, poor response to effective intervention, and the absence of exclusionary factors including intellectual disabilities, sensory deficits, serious emotional disturbance, lack of opportunity to learn, and low achievement resulting from limited proficiency in the language of instruction (Bradley et al., 2002). A strength of the model is that it specifies both low achievement and poor response to instruction as key criteria in addition to the typical exclusionary factors. There are several areas of needed research for the hybrid model however. One issue is the possibility that the two primary criteria of poor achievement and poor response to effective intervention are perfectly correlated except for measurement error. In a large-scale two-year longitudinal study of students beginning in first grade, measures of growth did not predict future reading achievement when current reading achievement was included as a predictor (Schatschneider et al., 2008). This would mean there are not really two different primary criteria. The second issue concerns several of the exclusionary criteria. Having intellectual disabilities would not seem to prevent some individuals from also having dyslexia, and the same would seem to be true for serious emotional disturbance. The origin of the hybrid model was a conference of educational researchers sponsored by the U.S. Department of Education Office of Special Education. Their focus is on special education services for students with disabilities. The rationale for excluding students with intellectual disabilities or serious emotional disturbance may be that they already are eligible for special education services because of these other disabilities. But it is not clear that teachers and others who provide services to students with intellectual disabilities or serious emotional disturbance are trained to identify and remediate academic-related disabilities such as dyslexia. Finally, there has yet to be a rigorous psychometric study of the reliability and validity of the model.

An expanded hybrid model proposed by Compton et al. (2006) added impaired phonological processing (sound matching and rapid naming) to the key criteria and used decision classification trees in an attempt to improve accuracy of identification. The addition of impaired phonological processing makes sense for identification of individuals with literacy-related learning disabilities such as dyslexia because a deficit in phonological processing is viewed as a primary cause of the vast majority of cases of dyslexia (but see Castles & Coltheart, 2004). Both the addition of the phonological processing criterion and decision classification trees improved the accuracy of the model for predicting poor reading in second grade.

A constellation model of dyslexia identification was proposed by Spencer et al. (2014). The model assumed that dyslexia is a latent construct that cannot be observed directly but its presence can be inferred by its effects on measurable indicators (Fletcher et al., 2019). The model distinguished causes (impaired phonological processing and genetic risk), consequences (impaired word and nonword reading, impaired reading comprehension relative to listening comprehension, and poor response to instruction and intervention), and correlates (attention deficit disorder, mathematics disorder, and sex) of dyslexia. Spencer et al. (2014) compared the longitudinal stability of a four-indicator model representing the four consequences specified by the model to alternative single-indicator model, finding substantially greater stability for the four-indicator version as would be expected. A rigorous empirical examination of the complete hybrid model has yet to be completed, however.

Pennington (2006) and colleagues (McGrath et al., 2020; Pennington et al., 2012) proposed the multiple deficit model of developmental disorders including dyslexia. Pennington et al. (2012) compared the fit of several single-deficit and multiple deficits models. The single-deficit models included deficits in either phonological awareness, naming speed, or language skill. Multiple-deficit models included a phonological core, multiple other deficit model and a multiple deficit, multiple predictor model. The results were that roughly equal numbers of participants fit the single deficit and multiple deficit models, but a phonological awareness deficit was the most common deficit for all models tested. Phonological deficits have been strongly associated with literacy impairment in other studies as well. For example, Catts et al. (2017) found that children with a phonological deficit in kindergarten were five times more likely to have dyslexia in second grade (see also Ramus et al., 2003; Saksida et al., 2016).

Probabilistic Decision Making

In addition to moving from single criterion to multiple criteria models, another approach for improving reliability and validity of identification is to move from a dichotomous categorization to a probabilistic outcome. Probabilistic models that acknowledge uncertainty used in other fields may have applicability in improving identification of individuals with dyslexia.

For example, if you have a physical exam that includes common laboratory testing of a blood sample, your physician can give you 10-year and lifetime probabilities of having a stroke or heart attack. Risk prediction is done using a risk calculator that is available online (http://tools.acc.org/ASCVD-Risk-Estimator/) so that if your physician does not provide these probabilities for you, you can enter your numbers and determine your probabilities yourself. The risk calculator uses demographic information (your age, sex, and race), historical information (history of diabetes or smoking), data obtained from your physical exam and labs (systolic blood pressure, diastolic blood pressure, total cholesterol, HDL cholesterol, LDL cholesterol) and current treatment regimens you are following (are you on hypertension treatment, on a statin, and on aspirin therapy?) to calculate risk. Importantly, the calculator can be used not only to predict current risk but to calculate the risk reduction obtained from treatments such as statin therapy. It also is being included as a dependent variable in randomized controlled trials of various heart attack and stroke reduction therapies. The risk prediction model underlying the calculator is based on a large-scale longitudinal study.

Bayes Theorem

Bayesian-based prediction models can be useful when informative priors in the form of known prevalence of a disorder exist and when the diagnostic data are not completely determinative. Bayes theorem begins with a prior probability that can be the prevalence of a disorder. The prior probability is updated using test results to end up with a posterior probability that takes both the prior probability and the test results into account.

Bayesian are used in medical diagnosis and policy considerations (Spiegelhalter, Abrams, & Myles, 2004). For example, consider the case of mammography and breast cancer. For a woman who is 40, the chance that she will have breast cancer in the next decade is small, fortunately, at 1.55 percent. If breast cancer is present, a positive mammogram will occur 75 percent of the time (e.g., true positive). If breast cancer is absent, a positive mammogram will still occur 10 percent of the time (e.g., false positive). Using the equation below to apply Bayes theorem to this problem requires three probabilities: the prior probability or base rate for the disease (x); the probability of a positive test result if the disease is present (y); and the probability of a positive test result if the disease is absent (z). For the example of a woman who is 40, x = .0155, y = .75, and z = .10.

xy/(xy+z1−x

Plugging these values into the equation gives the posterior probability of .11. This means that a woman who is 40 and gets a positive mammogram has only an 11% chance of actually having breast cancer. Following up a positive mammogram when breast cancer is not present carries a risk of complications. Because of Bayesian reasoning that takes the low prevalence of the disease for women in their 40s, it has been recommended that women in their 40s do not get routine annual mammograms unless they have a family history of breast cancer or other known risk factors. For a woman who is 70, the chances of having breast cancer in the next decade are higher at 4.09 percent. By changing x in the equation above to .0409, the posterior probability goes up to .24 meaning there is a 1 in 4 chance that breast cancer is present for a women in her 70s with a positive mammogram.

Wagner et al. (2019) provided an example of how Bayes theorem might be used to estimate the probability of the presence of dyslexia. Using the large-scale dataset from Spencer et al., (2014), dyslexia was operationally defined as scoring at or below the 5th percentile on a factor score representing the consequences of dyslexia according to the constellation model described previously (impaired word and nonword reading, impaired reading comprehension relative to listening comprehension, and poor response to instruction and intervention). Because the 5th percentile was chosen, the prior probability or chance of having dyslexia for this example was 5 percent. This prior probability can then be updated based on known risk factors. For example, boys are about 2 times more likely to have severe cases of dyslexia than girls are (Quinn & Wagner, 2015). Adding this to the model results in the chances of having dyslexia increasing from 5 to 7 percent if the individual is male and decreasing from 5 to 3 percent if the individual is female. Scoring at or below the 20th percentile on a battery of first-grade predictors increases the chances of having dyslexia from 5 to 15 percent. Co-occurring ADHD increases the chances of having dyslexia from 5 to 19 percent (Willcutt et al., 2007a, 2007b). Because family-risk of dyslexia increases an individual’s risk by approximately 4 times (Snowling & Melby-Lervåg, 2016), having an affected parent or sibling increases the chances from 5 to 26 percent. Combinations of risk factors, which commonly occur in real life, can be considered. For example, being male with ADHD increases the chances from 5 to 24 percent, and if an affected first-degree relative exists, the chances increase to 76 percent. Finally, if this same individual also scores low on the predictor battery, chances increase to 92 percent.

To obtain these probabilities, Bayes theorem was applied sequentially to data from the large-scale database used by Spencer et al. (2014), with additional risk estimates derived from the other cited studies. An important limitation of this example, however, is that it makes two assumptions that are unlikely to be true. First, sequentially applying Bayes theorem as was done to handle combinations of risk factors required the assumption that the risk factors were independent when they demonstrably are not. The second simplifying assumption was that the prevalence of dyslexia was 5 percent. This was an arbitrary estimate that does not reflect current thinking about the prevalence of dyslexia. The approach used in the present study does not make either problematical assumption.

The Prevalence of Dyslexia

A Bayesian approach to prediction works best when the prevalence of a disorder is known. Prevalence is used for the prior probability that is updated using data to obtain the posterior probability. Unfortunately, there has not been a consensus about the prevalence of dyslexia. Published prevalence estimates range from 3 to over 17 percent, with most falling below 10 percent (Hoeft et al., 2015; Snowling & Hulme, 2021). One reason for varying prevalence estimates is differences in the cut-points used for identification. A cut-off of 1.5 standard deviations below the mean typically result in prevalence estimates that fall in the range from 3 to 7 percent (Moll et al., 2014; Peterson & Pennington, 2012; Snowling & Melby-Lervag, 2016). Using a less stringent cut-off, Shaywitz et al. (1992) reported a prevalence of 17.4 percent of the school-age population. The operational definition that produced this higher prevalence was scoring at or below the 25th percentile in reading and/or an IQ-achievement regression-based definition of 1.5 standard deviations.

There is consensus that prevalence estimates vary depending on the cut-points used for identification. But this fact has not been incorporated in how prevalence is estimated; rather it has only been used to explain why there is so much variability in prevalence estimates in the literature. Recently, a new approach to estimating prevalence has been proposed that describes prevalence in terms of a distribution of prevalence values that vary as a function of severity as opposed to any single number (Wagner et al. 2019, 2020). For this approach, dyslexia was operationally defined as being worse at reading comprehension than at listening comprehension. If you have difficulty reading the words on the page, listening comprehension should be better than reading comprehension. Although it is clear that no single criterion is sufficient for reliable identification at the level of the individual, reading comprehension worse than listening comprehension was considered a reasonable proxy for estimating the prevalence of dyslexia at the population level.

Model-based meta-analysis was used to create simulated datasets that contained both listening and reading comprehension. The goal of traditional meta-analysis is to collect the information required to calculate an effect size from each study that was retained, and oftentimes to also code potential moderators that account for variability in effect sizes across studies. In contrast, the goal of model-based meta-analysis is to extract not an effect size from each study but rather a set of effect sizes that are typically represented by correlation matrices in studies of associations among multiple variables (Becker & Aloe, 2019; Cheung, 2015; Cheung & Chan, 2005). The correlation matrices are combined to create a composite correlation matrix that in turn can serve as data for subsequent modeling.

Wagner et al. (2019, 2020) used a composite correlation matrix to create a simulated dataset that contained both listening and reading comprehension among other variables. This made it possible to compute a new variable that was the difference between listening and reading comprehension. The distribution of this variable then represented the distribution of the prevalence of dyslexia in which prevalence values in the distribution varied as a function of severity. Examining this distribution resulted in three key findings that were replicated across the two studies. First, the prevalence of dyslexia can be represented as a normal distribution that varies as a function of severity as opposed to any single-point estimate. Second, typical samples of poor readers will contain more expected poor readers than unexpected poor readers. Third, cases with reading comprehension substantially worse than listening comprehension occur across the reading performance spectrum as opposed to occurring only in the lower tail of reading performance.

The Present Study

The goal of the present study was to apply Bayesian probabilistic models to predicting dyslexia in a way that eliminated the problematical assumptions required when Bayes theorem was applied sequentially in the example from Wagner et al. (2019) described previously. In particular, the study was designed to address the following six issues:

  1. Do commonly proposed multiple indicators of dyslexia make independent contributions to prediction as opposed to having predictive relations that are largely redundant with one another?

  2. Do family risk and response to instruction make contributions to prediction that are independent of typical test-score based predictors? Even when studies of dyslexia rely on multiple indicator models, the indicators typically are restricted to test scores representing common predictors of reading. In contrast, response to intervention, which cannot be measured by a single test score, is used for identification of individuals with dyslexia but with few exceptions (e.g., Erbeli et al., 2018; Spencer et al. 2014) is not included in empirical studies of multiple indicator models that include additional test scores. Family risk of dyslexia also appears to be an important predictor yet with few exceptions (e.g., Puolakanaho et al., 2007; Thompson et al., 2015) has not been included in studies of multiple indicator predictive models. For these reasons, we included both common predictors of reading that are represented by test scores as well as measures of response to intervention and family risk.

  3. How might recent advances in our understanding of the prevalence of dyslexia as a distribution of values rather than a single number be incorporated in Bayesian prediction models? The applications of Bayes theorem described earlier required having a single number representing prevalence. However, modern computational approaches for obtaining parameter estimates for Bayesian models use Markov Chain Monte Carlo estimation in which a distribution of prevalence values can be sampled from repeatedly.

  4. How do results for predicting unexpectedly poor reading compare to those for predicting simple poor reading. Although the concept of unexpectedness is a part of nearly all conceptualizations of dyslexia (Grigorenko et al, 2020), many studies that purport to target individuals with dyslexia use simple low achievement for identification. Samples of poor readers represent a mixture of individuals whose reading performance is low but consistent with their performance on a variety of measures of oral language and poor readers whose level of reading is unexpected based on their level of oral language performance (Wagner et al., 2020). For this reason, we wanted to compare prediction of simple poor reading with measures of unexpectedly poor reading.

  5. Some studies of individuals with dyslexia use poor reading comprehension as an important criterion whereas as others focus specifically on poor decoding. For that reason, we examined prediction for both poor reading comprehension and poor decoding (Aaron, 1991; Badian, 1999; Beford-Fuell, Geiger, Moyse, & Turner, 1995; Erbeli, Hart, Wagner, & Taylor, 2018; Fletcher et al., 2019; Spring & French, 1990; Stanovich, 1991).

  6. Because of potential differences in accuracy for the decision that an individual has dyslexia relative to the decision that dyslexia is not present, we examined accuracy for these decisions separately. And because this phenomenon may interact with severity, with more stringent cut-offs resulting in positive cases necessarily being closer to the cut-off than less stringent cut-offs (Waesche et al., 2011), we compared prediction accuracy at two levels of severity, the 20th and 5th percentiles.

Method

Model-based meta-analysis was used to produce a composite correlation matrix of key predictors of dyslexia. A simulated dataset was then created based on the composite correlation matrix. Three meta-analyses were used to obtain correlations for building the composite correlation matrix.

  1. Wagner et al. (2021) used model-based meta-analysis of correlation matrices obtained from nationally normed standardized tests to create a composite matrix for the variables reading comprehension, listening comprehension, and decoding. Although not analyzed in the study, phonological awareness was also available for inclusion. The initial search yielded 91 tests, 6 of which remained after applying six inclusionary criteria: (1) norm referenced; (2) nationally representative norming sample; (3) in English; (4) included subtests measuring listening comprehension, reading comprehension, vocabulary, decoding, and phonological awareness; (5) correlation matrix of subtests and subtest reliability available; and (6) included data from multiple ages or grades. The combined normative samples totaled 70,978 individuals. The model-based meta-analysis was carried out using the ‘metaSEM’ package in R (Cheung, 2015). metaSEM is a two-stage, structural equation modeling approach. The first stage consists of generating a composite correlation matrix from the studies included in the meta-analysis. The second stage involves modeling using the composite correlation matrix as data. For present purposes, we obtained the composite correlation matrix generated from the first stage using a multivariate random effects model. Syntax for using metaSEM to create the composite correlation matrix is presented in the Supplemental Methods section.

  2. Snowling and Melby-Lervag (2016) reported a meta-analysis of studies that compared children with family risk of reading disorders to controls not at risk. The meta-analysis included 95 studies based on 21 independent samples. Mean Cohen’s d effect sizes from Table 5 were obtained for vocabulary knowledge, phoneme awareness, nonword decoding, and reading comprehension. The mean Cohen’s d values were then converted to correlations.

  3. Stuebing et al. (2015) reported a meta-analysis of predictors of response to intervention. The meta-analysis included 28 studies comprising 39 samples. Meta-analytic mean correlations between growth in response to intervention and oral language, phonological awareness, reading comprehension, and word reading were obtained from Table 3 of this study.

Table 5.

Contingency Tables for Poor Decoding (DECODE) and Unexpectedly Poor Decoding (RESDEC) at Two Levels of Severity

Observed Predicted Percentage Correct
DECODE20 0 1
0 764 36 95.5
1 124 76 38.0
Overall Percentage Correct 84.0
DECODE5 0 1
0 947 3 99.7
1 43 7 14.0
Overall Percentage Correct 95.4
RESDEC20 0 1
0 784 16 98.0
1 180 20 10.0
Overall Percentage Correct 80.4
RESDEC5 0 1
0 949 1 99.9
1 50 0 0.0
Overall Percentage Correct 94.9
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Note. DECODE20 = decoding at or below the 20th percentile. DECODE5 = decoding at or below the 5th percentile. RESDEC20 = unexpectedly poor decoding at or below the 20th percentile of the distribution of the residuals from regression decoding on oral language. RESDEC5 = unexpectedly poor decoding at or below the 5th percentile of the distribution of the residuals.

Table 3.

Bayesian Logistic Regression Results Predicting Poor Decoding (DECODE) and Unexpectedly Poor Decoding (RDEC) at the 20th and 5th Percentiles

Poor Decoding (DECODE)
20th Percentile 5th Percentile
Predictor EAP 95% HPDI ESS EAP 95% HPDI ESS
Intercept −1.98* −2.22, −1.75 8449 −4.46* −5.11, −3.84 3481
OL −0.53* −0.74, −0.31 10772 −0.73* −1.11, −0.36 6176
PA .0.70* −0.94, −0.48 10005 −1.04* −1.47, −0.62 4680
FR −0.60* −0.80, −0.40 10657 −0.57* −0.93, −0.22 8076
RTI −0.43* −0.62, −0.23 11293 −0.49* −0.83, −0.14 9464
Unexpectedly Poor Decoding (RESDEC)
20th Percentile 5th Percentile
Predictor EAP 95% HPDI ESS EAP 95% HPDI ESS
Intercept −1.64* −1.83, −1.45 10425 −3.59* −4.03, −3.15 6461
PA −0.44* −0.63, −0.25 11065 −0.62* −0.98, −0.28 7919
FR −0.60* −0.80, −0.41 11582 −0.68* −1.00, −0.35 8007
RTI −0.26* −0.44, −0.09 12425 −0.35* −0.65, −0.04 10437
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Note. EAP = expected a posteriori. 95% HPDI = 95th percentile highest posterior density interval. ESS = effective sample size. RDEC = residual from regressing DECODE on oral language. OL = oral language. PA = phonological awareness. FR = family risk. RTI = response to intervention.

*

Bayesian analog to p < .05

Across the three meta-analyses, there were 153 independent effects and over 70,000 participants. The composite correlation matrix that resulted from combining results from the three meta-analyses included the variables reading comprehension, listening comprehension, decoding, phonological awareness, family risk, and response to treatment. No correlation was available between family risk and response to intervention from the combined meta-analyses. There are a number of ways to impute missing values in composite correlation matrices (Jak & Cheung, 2018). We used the approach based on the theory of convex optimization and semidefinite programming (Olvera Astivia, 2021). The resultant imputed correlation of .209 was included in the composite correlation matrix.

Next, the composite correlation matrix was used to create a simulated dataset containing 1,000 cases. The number of cases in the simulated dataset was chosen to provide a sample that faithfully reproduced the population parameters from the composite correlation matrix. The simulated dataset was created using a syntax script in SPSS version 25 software (IBM Corp., 2017). The syntax used is provided in the Supplemental Methods section. The correlations observed in the simulated dataset matched those from the composite correlation matrix, confirming the accuracy of the simulation. Finally, two additional variables representing unexpectedly poor reading comprehension and unexpectedly poor decoding were derived from the correlations in the composite correlation matrix by regressing both reading comprehension and decoding on the oral language variable.

Results

The composite correlation matrix, derived from the meta-analysis, and the descriptive statistics of the simulated data are presented in Table 1. The correlations were moderate with a few exceptions. The correlations between oral language and the residuals of reading comprehension regressed on oral language and decoding regressed on oral language were zero as expected. The fact that the means of all variables were 0 reflects the fact that the simulated data were based on correlations assuming a mean of zero and a standard deviation of 1, with the exception of the standard deviations for the two variables that represented residuals.

Table 1.

Composite Correlation Matrix

1. RC 1
2. OL .588 1
3. DECODE .565 .412 1
4. PA .544 .456 .526 1
5. FR .358 .273 .489 .326 1
6. RTI .680 .190 .340 .310 .209 1
7. RESRC .809 .000 .395 .341 .244 .703 1
8. RESDEC .351 .000 .908 .369 .413 .287 .434 1
1. 2. 3. 4. 5. 6. 7. 8.
Mean 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Standard Deviation 1.000 1.000 1.000 1.000 1.000 1.000 0.809 0.908
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Note. RC = reading comprehension. OL = oral language. DECODE = decoding. PA = phonological awareness. FR = family risk. RTI = response to intervention. RESRCS = residual from regressing RC on OL. RESDEC = residual from regressing DECODE on OL.

Bayesian logistic regressions were performed on the simulated data using Jags version 4.3.0. JAGS (just another Gibbs sampler) is a program for estimating parameters for Bayesian models using Markov Chain Monte Carlo (MCMC). A sample script for doing this is presented in the Supplemental Methods section.

The Bayesian logistic regression results are presented in Tables 2 and 3. Three key statistics are reported for each parameter estimate. The EAP or expected a posteriori represents the mean of the posterior distribution. A posterior distribution is created by repeatedly sampling from the prior distribution, updating the prior using the data, and ending up with a posterior distribution. The EAP is analogous to a logistic regression coefficient. Whereas EAP is a measure of central tendency, the 95% HPDI or highest posterior density interval is an estimate of dispersion of the scores from the measure of central tendency. Specifically, the 95% HPDI is the 95% interval of posterior values for which every point within the interval has a higher probability that any point outside the interval (Kaplan, 2014). HPDIs that do not contain 0 would correspond to an EAP that would be considered to be significant in frequentist statistics language. Finally, ESS or effective sample size is the sample size of a completely non-autocorrelated chain, an index of how much independent information there is in autocorrelated chains (Kruschke, 2015). It can be thought of as how many independent draws from the posterior distribution the Markov chain is equivalent to.

Table 2.

Bayesian Logistic Regression Results Predicting Poor Reading Comprehension (RC) and Unexpectedly Poor Reading Comprehension (RESRC) at the 20th and 5th Percentiles

Poor Reading Comprehension (RC)
20th Percentile 5th Percentile
Predictor EAP 95% HPDI ESS EAP 95% HPDI ESS
Intercept −2.93* −3.31, −2.55 4054 −6.22* −7.31, −5.17 1565
OL −1.25* −1.55, −0.94 6240 −1.20* −1.68, −0.70 3400
DECODE −0.60* −0.90, −0.31 8313 −0.47 −1.00, 0.09 3920
PA −0.42* −0.70, −0.15 9515 −0.64* −1.16, −0.11 4495
FR −0.19 −0.43, 0.06 10556 −0.26 −0.68, 0.19 6838
RTI −1.74* −2.06, −1.43 5054 −2.08* −2.63, −1.52 1829
Unexpectedly Poor Reading Comprehension (RESRC)
20th Percentile 5th Percentile
Predictor EAP 95% HPDI ESS EAP 95% HPDI ESS
Intercept −2.25* −2.53, −1.98 6623 −4.66* −5.37, −4.01 3049
DECODE −0.45* −0.70, −0.20 7933 −0.23 −0.67, 0.24 6807
PA −0.22 −0.45, 0.02 9755 0.01 −0.42, 0.42 8068
FR −0.07 −0.30, 0.14 10017 −0.28 −0.66, 0.10 8543
RTI −1.68* −1.94, −1.40 6686 −1.98* −2.42, −1.51 3095
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Note. EAP = expected a posteriori. 95% HPDI = 95th percentile highest posterior density interval. ESS = effective sample size. RC = reading comprehension. OL = oral language. RRC = residual from regressing RC on OL. DECODE = decoding. PA = phonological awareness. FR = family risk. RTI = response to intervention.

*

Bayesian analog to p < .05

The results for predicting poor reading comprehension and at the 20th and 5th percentiles using oral language, decoding, phonological awareness, family risk, and response to intervention are presented in the top half of Table 2. Beginning with poor reading comprehension at or below the 20th percentile, each of the predictors makes an independent contribution to prediction with the exception of family risk. The largest contribution to prediction was response to intervention, following by oral language, decoding, and phonological awareness in that order. The pattern of results for prediction of poor reading comprehension at or below the 5th percentile was similar with the exception that neither decoding nor family risk made an independent contribution to prediction.

Turning to unexpectedly poor reading comprehension, the results for predicting unexpectedly poor reading comprehension at the 20th percentile and 5th percentiles are presented in the bottom half of Table 2. For predicting unexpectedly poor reading comprehension at or below the 20th percentile, response to instruction remained the strongest predictor followed by decoding. Neither family risk nor phonological awareness contributed to prediction. Moving to unexpectedly poor reading comprehension at the 5th percentile, only response to intervention made an independent contribution to prediction.

Moving from reading comprehension to decoding, results from predicting poor decoding at the 20th and 5th percentiles are presented in the top half of Table 3. The predictors were oral language, phonological awareness, family risk, and response to intervention. For poor decoding at the 20th and 5th percentiles, every predictor made independent contributions to prediction. For poor decoding, phonological awareness was the strongest predictor at both levels of severity. For poor decoding at the 20th percentile, phonological awareness was followed in order by family risk, oral language, and response to intervention. For poor decoding at the 5th percentile, oral language edged out family risk for the second strongest predictor.

Turning to unexpectedly poor decoding, results of which are presented in the bottom half of Table 3, at both the 20th and 5th percentiles, family risk was the strongest predictor followed by phonological awareness and response to intervention.

Contingency tables for predicting poor reading comprehension and unexpectedly poor reading comprehension are presented in Table 4, and contingency tables for predicting poor decoding and unexpectedly poor decoding are presented in Table 5. Although overall percentage correct values are good to excellent for all of the models, there is a clear difference in accuracy for predicting positive cases of dyslexia as opposed to negative cases. This difference in accuracy of prediction increases with increasing severity of the reading problem, yet overall percentage correct remains strong or even increases because there are more negative cases relative to positive cases moving from the 20th to the 5th percentiles.

Table 4.

Contingency Tables for Poor Reading Comprehension (RC) and Unexpectedly Poor Reading Comprehension (RC) at Two Levels of Severity

Observed Predicted Percentage Correct
RC20 0 1
0 758 42 94.8
1 76 124 62.0
Overall Percentage Correct 88.2
RC5 0 1
0 945 5 99.5
1 29 21 42.0
Overall Percentage Correct 96.0
RESRC20 0 1
0 754 46 94.3
1 112 88 44.0
Overall Percentage Correct 84.2
RESRC5 0 1
0 944 6 99.4
1 37 13 26.0
Overall Percentage Correct 95.7
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Note. RC20 = reading comprehension at or below the 20th percentile. RC5 = reading comprehension at or below the 5th percentile. RESRC20 = unexpectedly poor reading comprehension at or below the 20th percentile of the distribution of the residuals from regression reading comprehension on oral language. RESRC5 = unexpectedly poor reading comprehension at or below the 5th percentile of the distribution of the residuals.

Classification statistics derived from the contingency tables are presented in Table 6. Percent correct is relatively high for all of the models. However, as reflected in the contingency tables, specificity (i.e., percentage of negative cases who are accurately classified) is considerably higher than sensitivity (i.e., percentage of positive cases who are accurately classified). These results are mirrored in the positive predictive value and negative predictive value columns. In general, classification accuracy appears to be better for impaired comprehension than for impaired decoding, and better for simple low achievement than for unexpected or residualized poor achievement, as reflected in the sensitivity, positive predictive value, and fscore values, and to a lesser extent in the area under the curve values. When classification is based on the known probability of group membership, sensitivity improves markedly but at the expense of specificity. Because fscore includes both positive predictive value and sensitivity, the values to not change much when classification models switch from assuming .5 probability of group membership to the actual probabilities of .2 and .05.

Table 6.

Classification Statistics Calculated from Contingency Tables Resulting from the Bayesian Logistic Regression Analyses

DV % Correct Sensitivity Specificity PPV NPV AUC fscore Cut Off Sensitivity at prob Specificity at prob fscore at prob
RC20 88.2% 62.0% 94.8% 74.7% 90.9% .93 .68 .26 97.5% 61.05% .69
RC5 96.0% 42.0% 99.5% 80.8% 97.0% .96 .55 .04 92.0% 86.3% .41
RESRC20 84.2% 44.0% 94.3% 65.7% 87.1% .87 .53 .20 98.0% 44.2% .61
RESRC5 95.7% 26.0% 99.4% 68.4% 95.7% .91 .38 .05 80.0% 80.6% .30
DEC20 84.0% 38.0% 95.5% 67.9% 86.0% .82 .49 .29 96.5% 33.0% .51
DEC5 95.4% 14.0% 99.9% 70.0% 95.7% .88 .26 .06 80.0% 81.3% .30
RESDEC20 80.4% 10.0% 98.0% 55.6% 81.3% .74 .18 .25 67.5% 68.2% .46
RESDEC5 94.9% 2.0% 99.9% 49.5% 94.9% .78 .04 .04 68.0% 71.8% .19
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Note. % correct is percentage of correct classification. Sensitivity is the percentage of individuals with a condition who test positive. Specificity is the percentage of individuals who do not have a condition who test negative. PPV or positive predictive value is the percentage of people who test positive who actually have the condition. NPV or negative predictive value is the percentage of people who test negative who do not actually have the condition. AUC is area under the curve. Fscore is the harmonic mean of positive predictive value and sensitivity. Cut off is the empirically derived probability based on sensitivity and specificity. Sensitivity at prob is the value of sensitivity if you used the probability of group membership (.2 or .05) as the cutoff for classification. Specificity at prob is the value of sensitivity if you used the probability of group membership (.2 or .05) as the cutoff for classification. Fscore at prob is the harmonic mean of positive predictive value and sensitivity if you used the probability of group membership (.2 or .05) as the cutoff for classification. RC20 = reading comprehension at or below the 20th percentile. RC5 = reading comprehension at or below the 5th percentile. RESRC20 = unexpectedly poor reading comprehension at or below the 20th percentile of the distribution of the residuals from regression reading comprehension on oral language. RESRC5 = unexpectedly poor reading comprehension at or below the 5th percentile of the distribution of the residuals. DECODE20 = decoding at or below the 20th percentile. DECODE5 = decoding at or below the 5th percentile. RESDEC20 = unexpectedly poor decoding at or below the 20th percentile of the distribution of the residuals from regression decoding on oral language. RESDEC5 = unexpectedly poor decoding at or below the 5th percentile of the distribution of the residuals.

Discussion

The results support using model-based meta-analysis as a way to provide estimates of population correlations between key constructs. Results from the Bayesian logistic regression prediction provide strong support for predictive models of dyslexia that are based on multiple criteria. For nearly all of the models, independent contributions to prediction were found for multiple if not all predictors.

The constructs of response to intervention and family risk are commonly believed to be predictors of reading problems but rarely have they been competed with other common predictors of reading problems such as phonological awareness, decoding, and oral language. Because of this, there is a dearth of knowledge about whether their contributions to prediction are independent of or largely redundant with common, test-based predictors. Response to intervention was the strongest predictor of poor reading comprehension and unexpectedly poor reading comprehension, making an independent contribution to prediction even with the other common predictors in the model. Family risk for dyslexia was the second strongest predictor of poor decoding, and the strongest predictor of unexpectedly poor decoding, even with other common predictors in the model. This means, for example, that the power of family risk for predicting unexpectedly poor decoding is not accounted for by its relations with other common predictors such as phonological awareness that were included in the model. In the future, it will be important to incorporate additional predictors such as genome-wide polygenic scores representing genetic influences (Selzam et al., 2017) and variables derived from imaging when they are at sufficient levels of reliability for prediction at the level of the individual. It also will be important to look more closely at differences in prediction across age groups.

Regarding implications for theory, our study was not designed to test specific alternative models of the nature of dyslexia. However, the fact that multiple predictors from different levels of analysis (e.g., test scores, response to instruction and intervention, and family risk) made independent and substantial contribution to prediction provides support for multi-factor models of dyslexia that include variables from different levels of analysis (e.g., Erbeli et al., 2018; McGrath et al., 2020; Pennington, 2006; Pennington et al., 2012; Spencer et al., 2014). A previous argument for multiple indicator models was potentially improved reliability, and this is indeed an important advantage. However, if all of the predictors in a model are identically related to the criterion of having dyslexia but each predictor is only moderately reliable, the expected results would be a substantial logistic regression coefficient for the predictor that happened to be most related to the criterion for that sample and virtually no substantial coefficients for other predictors. Our results therefore support the idea that the construct of dyslexia itself is multifaceted, with multiple predictors required not only for reliable identification but also for valid coverage of the construct.

Regarding how prediction might be improved, a promising avenue is to identify additional relevant predictors that were not included in the present study. One source of additional predictors is the presence of co-occurring conditions. For example, a disorder in mathematics is a commonly observed co-occurring condition (Moll et al., 2020; Wilcutt et al., 2007, 2013). Daucourt et al. (2020) reported a meta-analysis of 38 genetically informative studies (i.e., twin studies) of third through ninth-grade students. The genetic, shared environmental, and unshared environmental correlations between reading and math were large, with values of .71, .90, and .56 respectively. Because Daucourt et al. (2020) also included ADHD in their meta-analysis of twin studies, it was possible to compare the relative magnitudes of the estimated genetic, shared environmental, and unshared environmental correlations between reading and mathematics and reading and ADHD. For reading and ADHD, the estimated genetic, shared environmental, and unshared environmental correlations of .42, .64, and .20, respectively, were substantially less than the respective values of .71, .90, and .56 for reading and mathematics. They concluded that reading and math may share more domain-general risk factors than reading and ADHD symptoms. A second meta-analysis (Joyner & Wagner, 2020) of 36 studies that addressed comorbidity between reading and mathematics disorder provided similar findings. Students with a mathematical disability were just over two times more likely to also have a reading disability.

Regarding improving prediction of unexpectedly poor reading in particular, one problem is that residual scores incorporate measurement error of both variables that are used to create the residual. However, it is possible that prediction of unexpectedly poor reading can be improved by using different predictors. Across three studies that included a model-based meta-analysis and two empirical studies, Wagner and Lonigan (2022) found that predictors that measured unexpectedly poor performance on the predictors (e.g., poor phonological awareness compared to expectations based on vocabulary) were stronger predictors of unexpectedly poor reading than were predictors that represented absolute poor performance.

Finally, there are advantages to modeling the criterion of reading comprehension or decoding as a continuous variable instead of as a dichotomous variable (MacCallum et al., 2002). We did not do that here because our focus was identification of cases of relative or absolutely poor performance. However, with our existing data, modeling the dependent variables as continuous rather than dichotomous outcomes would merely require switching from Bayesian logistic regression models to Bayesian multiple regression models.

Regarding implications for practice, the results support supplementing test scores with an investigation into family history (Puolakanaho et al., 2007; Thompson et al., 2015) and an analysis of evidence about response to instruction and intervention (Fletcher et al., 2019; Stuebing et al., 2015). How best to determine a family history that is predictive of dyslexia and how best to determine adequacy of response to instruction and intervention—particularly from historical data—are important areas of additional research (Fuchs et al. 2004). The results do provide strong support for multifactor models of identification, particularly models that include but are not limited to constructs that are represented by test scores.

The finding that prediction models are more accurate for predicting the absence of a condition like developmental dyslexia compared to predicting that it is present provides indirect support for a multi-tiered systems of support (MTSS) model and potentially a change in philosophy about screening and identification. One feature about Bayesian prediction models is the attention given to uncertainty. In the present context, one can determine that an individual does not have dyslexia with more certainty that one can determine that dyslexia is present. In the context of an MTSS model, it makes sense to be overly inclusive in providing extra help to potentially at-risk students and to withdraw extra help after a child provides clear evidence that a genuine reading problem does not exist (Fletcher et al., 2015). In the context of screening and identification, identifying students who are not at-risk can be done with considerably more certainty than identifying students who are at risk for or have dyslexia.

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1

Acknowledgments

The research described in this article was supported by Grant Number P50 HD52120 from the Eunice Kennedy Shriver National Institute of Child Health and Human Development.

Footnotes

The authors have no conflicts of interest to disclose.

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