The children

Eight children were recruited by public announcement to area schools, daycare facilities, and early childhood programs. To participate, children had to be identified as having PD, characterized by a reduced consonantal inventory relative to English and to expected phonological development. This was established by performance at least one standard deviation below the normative mean reported for age- and gender-matched peers on the Goldman-Fristoe Test of Articulation–2 (Goldman & Fristoe, Reference Goldman and Fristoe2000), with a minimum of six sounds produced in error across contexts on this measure. Children also had to be within three and six years of age because this is an approximate time frame for accelerated speech sound normalization (Shriberg et al., Reference Shriberg, Kwiatkowski and Gruber1994). Inclusionary criteria further required that children perform within typical limits on a battery of diagnostic tests (Gierut, Reference Gierut, Dinnsen and Gierut2008b: 42), which evaluated hearing acuity, oral-motor structure and function, non-verbal intelligence, working memory, expressive and receptive vocabulary, and language. Exclusionary criteria included literacy, bilingualism and concurrent enrollment in treatment for speech and/or language disorders. In all, three boys and five girls met these criteria; their mean age was 4 ; 0 (years; months) (range 3 ; 0–5 ; 5), shown in Table 1.

Table 1.

Participant and experimental characteristics

Note: GFTA-2=Goldman-Fristoe Test of Articulation–2 (Goldman & Fristoe, Reference Goldman and Fristoe2000) standard score.

Expressive phonological samples and analyses

Descriptions of the expressive phonologies of the children were developed. Production data were elicited using the Phonological Knowledge Protocol (PKP; Gierut, Reference Gierut, Dinnsen and Gierut2008b). This is a conventional structured probe that samples each English sound in each relevant word position in multiple exemplars. It provides for representative sampling of the statistical patterns of English by including, for example, words from dense/sparse neighborhoods, and frequent/infrequent words (Gierut, Reference Gierut, Dinnsen and Gierut2008b: 70–92 for statistics). It further provides for elicitation of minimal pairs and morphophonemic alternations as evidence of the phonemic status of sounds in a child's expressive phonology.

PKP data were collected using a spontaneous picture-naming task. A child's responses were digitally recorded by the experimenter, and subsequently transcribed by independent trained listeners using narrow notation of the IPA. Transcribers had no contact with the children or families, and were blinded to the experiment. Phonetically transcribed responses were entered into an archival database by a separate team of research assistants who were also blinded to the experiment. Archival entries were used to compute percentages of accurate production and to develop standard phonological analyses (Dinnsen, Reference Dinnsen, Elbert, Dinnsen and Weismer1984). Phonological analyses established each child's phonetic and phonemic inventories, distribution of sounds and rule-governed alternations.

The phonemic inventory was central to this study because it delineated which sounds were used by a child to differentiate meaning, and which were not. Sounds that were present in the phonemic inventory did not warrant treatment because they were already functional (i.e. contrastive) in a child's expressive phonology. By comparison, sounds excluded from the phonemic inventory were non-contrastive and produced with near 0% accuracy on the PKP (Gierut & O'Connor, Reference Gierut and O'Connor2002). These warranted treatment to bring a child's expressive phonology on track with typical development and in line with the English language.

Linguistically, sounds excluded from the phonemic inventory reflect phonotactic constraints on the grammar (Dinnsen, Reference Dinnsen, Elbert, Dinnsen and Weismer1984: 12). There are three general types of phonotactic constraints: inventory constraints, which ban specific sounds from occurring (e.g. velar fricatives are prohibited in English); positional constraints, which ban specific sounds from occurring in particular contexts (e.g. velar nasals occur word-finally, but are prohibited word-initially in English); and sequence constraints, which ban certain sequences of sounds (e.g. consecutive sonorant–sonorant consonants such as /mr-/ are prohibited onsets of English). In this study, inventory constraints informed the independent variable because sounds excluded from the phonemic inventory were the focus of treatment for all children. Likewise, inventory constraints were measured as the dependent variable to establish which sounds excluded from the inventory evidenced gains in accuracy of production. On average, children of the study excluded ten sounds from the phonemic inventory (range 7–16; Table 1).

Experimental design

A single-subject, staggered, multiple-baseline (MBL) across-subjects design was used. Single-subject designs are based on the premise that a child serves as his or her own control (McReynolds & Kearns, Reference McReynolds and Kearns1983). By their very nature, single-subject designs take into account potential individual differences associated with internal or external factors, e.g. age, severity of PD, attention or family dynamics. The MBL is one particular single-subject design that is implemented as a period of no treatment followed by treatment for each child. In a staggered application of the MBL, children are enrolled in treatment sequentially, rather than concurrently. In an across-subjects application of the MBL, each child serves as one leg of a given experimental condition, such that effects are culled from multiple children, rather than multiple behaviors. Legs of the MBL further provide for replication as a reflection of the generalizability of treatment effects. For phonological treatment, generalizability is typically achieved by demonstrating that the effects of treatment are not sound-specific.

As applied in this study, the no-treatment or baseline phase of the MBL ensured stability of a child's performance prior to the instatement of treatment, namely, that sounds excluded from the phonemic inventory showed little fluctuation in production accuracy (±10% variation, following McReynolds & Kearns, Reference McReynolds and Kearns1983). To assess stability of performance over time, each child was given more than one baseline (Table 1). To assess stability across legs of the MBL, the number of baselines was increased by one as successive children enrolled. Based on principles of single-subject design, baseline stability helps to establish a causal relationship between experimental treatment and change in performance. The ongoing stability of time-lagged baselines further demonstrates that change in performance is not attributable to maturation. If maturation were at work, then children next to enroll would be expected to evidence gains in the absence of treatment because the extended period of baseline monitoring would have afforded them the time to mature.

As applied herein, the treatment phase of the MBL supplied the independent variable, namely, treatment of a sound excluded from a child's phonemic inventory using stimulus words that orthogonally varied in density and frequency. Four children were taught a sound in words from dense neighborhoods, and four others in words from sparse neighborhoods. Within and across conditions, frequency was further varied, such that two of four children were treated on frequent, and two others on infrequent words. Hereafter, we refer to the conditions as Dense–Freq, Dense–Infreq, Sparse–Freq and Sparse–Infreq. Through these manipulations, it was possible to examine the blocked effects of dense/sparse neighborhoods and frequent/infrequent words as in prior studies, in addition to the combined effects of density and frequency in keeping with the central aim of this study.

The dependent variable was generalization, operationalized as the percentage accuracy of production of sounds excluded from a child's baseline phonemic inventory as measured on the PKP. Generalization data were collected longitudinally, with samples obtained at baseline, at the completion of each treatment step (outlined below), and immediately following completion of treatment. Generalization data were also collected throughout treatment on a variable ratio schedule averaging two sessions. On average, 13 PKP samples (range: 8–16) were obtained from each child. Consistent with the enrollment procedures above, each PKP sample was digitally recorded and phonetically transcribed, with transcriptions entered into the archival database. The data were used to compute separate percentages of accuracy for each sound excluded from a given child's phonemic inventory at each sampling point in time. Thus, accuracy was documented for every sound excluded from the phonemic inventory of a child at approximately thirteen points in time, corresponding to the thirteen PKP samples that were administered on average. The full set of longitudinal data was then evaluated in data analysis.

Because accuracy data were derived from descriptive phonetic transcriptions, it was necessary to document reliability. Transcription reliability was established for 10% of the longitudinal PKP data collected from each child. Independent transcriptions from two judges were compared point-to-point, with mean agreement being 92% based on 2,317 consonants transcribed.

Treated stimuli

The overarching goal of treatment was the elimination of inventory constraints as the unified linguistic characterization of all children's error patterns. Each child was taught one sound excluded from his or her phonemic inventory. The treated sound was pseudo-randomly assigned to accommodate individual differences in children's inventories while also providing for replication. Half the children were taught a fricative, and half a liquid, balanced by condition. The treated fricative was either /f/ or /s/, and the liquid was either /l/ or /r/; these are shown in Table 1 by condition and by child. The treated sound was taught in the initial position of eight treated words that conformed to the density/frequency characteristics of a child's experimental assignment; the treated words are shown in the ‘Appendix’. Treated words were picturable, of suitable content for a child, and non-overlapping with PKP items so as to keep treated stimuli distinct from generalization stimuli. Several considerations went into the selection of the treated sounds and treated words, as detailed below.

Treated words

Treated words were selected based on density and frequency values, as retrieved from the Hoosier Mental Lexicon (Nusbaum et al., Reference Nusbaum, Pisoni and Davis1984). This is an online database that supplies the density of 20,000 words of the Merriam-Webster Dictionary, with corresponding frequencies from Kučera and Francis (Reference Kučera and Francis1967) . An adult corpus was used for consistency with the two prior treatment studies involving the manipulation of density and frequency in PD (Gierut et al., Reference Gierut, Morrisette and Champion1999; Morrisette & Gierut, Reference Morrisette and Gierut2002), which the present study aimed to extend and replicate. Some have suggested that child corpora may provide a more accurate reference (Dollaghan, Reference Dollaghan1994). Yet others argue that adult corpora may be extended to the study of children for reasons associated with the normalization of standard distributions (Kelly & Martin, Reference Kelly, Martin, Gleitman and Landau1994), correlations showing the equivalence of corpora (Storkel & Hoover, Reference Storkel and Hoover2010a) and behavioral demonstrations of similar effects across corpora (Gierut & Dale, Reference Gierut and Dale2007). This debate notwithstanding, an adult corpus was used herein for ease of translation and uniformity across studies of PD, and to enable future comparisons between the performance of children and adults as a means of examining developmental continuity.

To initiate stimulus selection, conventional operational definitions based on the raw values of density and frequency (Luce, Reference Luce1986) were applied, such that words from dense neighborhoods were to have ten or more similarity neighbors, and frequent words were to have a raw count of 100 or more occurrences per million. In this study, treated words from dense neighborhoods had a mean of twenty-one neighbors, whereas those from sparse neighborhoods had a mean of four neighbors. Treated words that were classified as frequent had a mean of 293 occurrences and infrequent, twenty-one occurrences. In all, four sets of treated words were identified in accord with the crossing of density and frequency by experimental condition.

Because density and frequency are correlated with other properties of word structure, it was necessary to consider and control for alternate (and potentially confounding) variables in stimulus selection. Specifically, for density, there is an established negative correlation with word length (Pisoni, Nusbaum, Luce & Slowiaczek, Reference Pisoni, Nusbaum, Luce and Slowiaczek1985): dense neighborhoods are largely comprised of short words. Indeed, in this study, the treated words from dense neighborhoods averaged 3·2 segments, whereas those in sparse neighborhoods averaged 4·5 segments in length. This reflected the expected negative correlation between density and length (r(64)=−0·78, p<0·001), but it also resulted in a statistically significant difference in length between the dense/sparse stimulus sets (t(62)=−8·33, p<0·001). Typically, laboratory studies handle such correlations by holding length constant in stimulus selection; however, this may not be feasible, ecologically valid or necessarily desirable when naturalistic stimuli are used and when children are involved as participants (see Storkel, Reference Storkel2004a, for review). Such was the case in the present study. Accordingly, the recommended alternative (p. 1465) is to transform raw density values using z-score conversion and/or median transformation for length. Transformed density values are computed relative to all words that are of the same length in a given corpus (i.e. the Hoosier Mental Lexicon). In this way, length is taken into account to minimize potential confounds. Importantly, transformed density values appear to be better predictors of the behavioral patterns to emerge from experimental manipulations of density (p. 1466).

We followed this standard by applying both the procedure for density z-score conversion and density median transformation as outlined by Storkel (Reference Storkel2004a: 1458–60). Particular attention was given to the data provided about the Hoosier Mental Lexicon. Specifically, Storkel (Table 2, p. 1459) reported the mean and standard deviation for neighborhood density by word length for all words of the Hoosier Mental Lexicon; these values were used in computation of density z-scores. Likewise, Storkel (Table 2, p. 1459) reported the median and interquartile range for neighborhood density by word length for all words of the Hoosier Mental Lexicon; these values were used in computation of density median transformations.

Table 2.

Summary of blocked and orthogonal results based on the criterion cut-off of 10% mean generalization accuracy in coding yes/no generalization (Elbert et al., Reference Elbert, Dinnsen and Powell1984), the rank procedure for replicated AB design (Marascuilo & Serlin, Reference Marascuilo and Serlin1988), and effect size for single-subject design (Beeson & Robey, Reference Beeson and Robey2006)

¹ n=Number of factors evidencing positive gain; N=Number of possible factors examined for positive gain (i.e. number of subjects×treated and untreated sounds from the same and different manner classes; see Morrisette & Gierut, Reference Morrisette and Gierut2002: 150).

² Inadequate power due to the number of factors evidencing positive gain by children enrolled in these conditions.

With referent values in place, density z-scores were calculated herein using the equation: (Obtained raw density value – M)/SD (Storkel, Reference Storkel2004a: 1458). That is, for each treated word in each condition, the obtained raw density value was subtracted from the mean density value reported by Storkel (Table 2, p. 1459) for all words of the Hoosier Mental Lexicon that were of the same length. This value was then divided by the standard deviation, also reported by Storkel (Table 2, p. 1459) for all words of the Hoosier Mental Lexicon that were of the same length. To illustrate, funny was a treated word associated with the dense condition (see ‘Appendix’). It consists of four segments, and has a raw density value of eleven. The mean density of all four-segment words in the Hoosier Mental Lexicon is 6·87 (SD=4·53) (Table 2, p. 1459). Thus, the density z-score of funny was 0·91 (i.e. (11–6·87)/4·53). By comparison, lazy was a treated word associated with the sparse condition (see ‘Appendix’). It too consists of four segments, but has a raw density value of seven. Its density z-score was 0·03 (i.e. (7–6·87)/4·53). This procedure was followed to derive the mean density z-scores for treated words used in the dense/sparse conditions. The average density z-score for the dense condition was 0·83, and the sparse condition −0·09, resulting in a statistically significant difference in density by condition (t(62)=4·52, p<0·001). Importantly, there was no correlation between the density z-scores and word length (r(64)=−0·17, p=0·18).

Density median transformation scores were also computed, using the equation: (Obtained raw density value – Mdn)/1/2 IQR) (Storkel, Reference Storkel2004a: 1458). Procedures were much the same as above, with median transformation scores generated for each treated word used in the dense/sparse conditions. Values reported by Storkel (Table 2, p. 1459) again supplied the median and inter-quartile range for density based on all words of the Hoosier Mental Lexicon by length. The resulting mean density median transformation for treated words of the dense condition was 1·17, and for treated words of the sparse condition 0·18. Density median transformation values were statistically significant by condition (t(62)=3·72, p<0·001), with no correlation between the transformed density values and length (r(64)=−0·14, p=0·28).

Together, the density conversions provided the needed experimental safeguard to demonstrate that the treated stimuli were distinct in density by condition, even when length was taken into account. This established that density, not length, was a primary independent variable. (For completeness, the length of the treated words sorted by frequency was not statistically different (t(62)=−0·14, p=0·89). Average length of frequent versus infrequent treated words was 3·8 versus 3·9 segments, 2·8 versus 2·9 biphones and 1·4 versus 1·5 syllables, respectively.)

Besides length, density is positively correlated with phonotactic probability (Vitevitch & Luce, Reference Vitevitch and Luce2004): words in dense neighborhoods are largely comprised of common sounds and sound sequences. Thus, it was necessary to establish that treated words from dense/sparse neighborhoods did not differ in phonotactic probability as a spurious variable. We followed standard procedures recommended in the literature (Storkel, Reference Storkel2004a), computing the sums of segment and biphone frequencies for each stimulus word using a publicly available calculator (Vitevitch & Luce, Reference Vitevitch and Luce2004), with z-score conversion for length. Resulting values were binarily coded using the rubric that positive phonotactic values reflect common sound sequences, and negative values rare sound sequences. Chi-square analyses revealed no differences in commonality or rarity of the phonotactic patterns associated with treated stimuli from dense versus sparse neighborhoods. This held for sum of segment frequencies (χ² (1, N=64)=1·82, p=0·18) and sum of biphone frequencies (χ² (1, N=64)=0·58, p=0·45). The proportion of common:rare words was 1·2:1, yielding a nearly equivalent distribution of phonotactic probability by density. Density, not phonotactic probability, was thus retained as a primary independent variable.

Frequency is likewise correlated with other statistical cues. Specifically, there is a negative correlation between frequency and age-of-word-acquisition (AoA; Gilhooly & Watson, Reference Gilhooly and Watson1981). AoA is a reliable and behaviorally valid subjective estimate of when a given set of words was first acquired, as determined by child (Walley & Metsala, Reference Walley and Metsala1992) or adult judges (Gilhooly & Logie, Reference Gilhooly and Logie1980). The observed correlation is that high-frequency words tend to be earlier acquired, although the directionality of the effect remains a point of debate (Gilhooly & Watson, Reference Gilhooly and Watson1981). Once again, it was necessary to eliminate AoA as a potential confound, rendering frequency as a primary independent variable. This was accomplished by retrieving the AoA values of stimulus words in reference to the normative counts of Gilhooly and Logie (Reference Gilhooly and Logie1980) and Bird, Franklin and Howard (Reference Bird, Franklin and Howard2001) . Resulting AoA values were binarily coded, with words acquired before age six defined as early acquired in keeping with the inclusionary criteria for participation in this study. Chi-square analyses revealed no difference in the AoA values of frequent versus infrequent stimuli (χ² (1, N=44)=0, p=1·0). Overall, the ratio of early:late AoA words was 1·1:1, with near equal distribution of AoA by frequency. As above, frequency, not AoA, was established as a primary independent variable.

Imageability was a related consideration because it is negatively correlated with AoA (Bird et al., Reference Bird, Franklin and Howard2001): earlier acquired words tend to be highly imageable, keeping in mind too that the treated words used in this study were to be picturable. Above, we demonstrated that treated words were equivalent in AoA. It might be expected then that they would be comparable in imageability, given the reported correlation. Nonetheless, this needed to be confirmed. Subjective imageability ratings were retrieved from Bird et al. (Reference Bird, Franklin and Howard2001) for the treated words. Mean imageability ratings of treated words were 255 versus 213 for dense versus sparse neighborhoods, respectively, and 230 versus 259 for frequent versus infrequent stimuli, respectively. There was no significant difference in the imageability ratings of treated words sorted by density (t(28)=0·47, p=0·64), or sorted by frequency (t(28)=−0·29, p=0·78).

Beyond these controls, developmental phonological characteristics of the treated words were taken into account, given the interest in expressive phonology. The classification scheme of Shriberg and colleagues (Reference Shriberg, Kwiatkowski and Gruber1994) was used to code each sound of each treated word by its expected developmental order of emergence. This scheme sorts sounds into early-8 /m n p b d w j h/, mid-8 /ŋ k ɡ t f v ʧ ʤ/ and late-8 /θ ð s z ʃ Ʒ r l/ sets. The distribution of early-, mid- and late-8 segments and clusters in treated words from dense neighborhoods was 8, 17, 31 and 7, respectively; thus, treated words in dense neighborhoods were largely composed of late-8 sounds. For sparse neighborhoods, the distribution of early-, mid- and late-8 segments and clusters was 12, 17, 45 and 15, respectively; again, treated words in sparse neighborhoods were largely composed of late-8 sounds. Chi-square analyses showed no significant difference in the segmental and cluster composition of treated words based on the expected developmental order of emergence when sorted by density (χ² (3, N=152)=1·9, p=0·59). Likewise, there was no significant difference in the segmental composition of treated words based on developmental order when sorted by frequency (χ² (3, N=152)=0·19, p=0·98). As before, treated words sorted by frequency consisted mostly of late-8 sounds. These analyses ensured that the segmental make-up of treated words was comparable on developmental phonological grounds.

Two final phonological considerations were taken into account in treated word selection. To the extent possible, treated stimulus sets were comprised of unique words that were not also neighbors (minimal pairs) of each other: 94% of treated stimuli (60 of 64 words) were non-overlapping in phonological form. The intent was to constrain a child's exposure to the neighborhood structure of the treated words per se, without further enhancement of that structure in the form of minimal pair contrasts. Also, to the extent possible, treated stimuli did not form neighbors (minimal pairs) with PKP words: 98% of relevant PKP words (423 of 431 words) were non-overlapping in phonological form with treated words. The intent was to minimize the possible inflation of generalization effects by keeping the treated stimuli distinct from those used to measure generalization.

It must be recognized that any number of added factors might have been taken into account in stimulus selection, e.g. featural distinction, stress and grammatical or semantic class. Following Magnuson and colleagues (Reference Magnuson, Dixon, Tanenhaus and Aslin2007), these were left free to vary. Our objective was to capture the range of variation in English, while at the same time block for the primary variables of interest. In this regard, density and frequency were the key properties varied within a stimulus set as independent variables, while word length, phonotactic probability, AoA, imageability, developmental order of segmental emergence and minimal pairs were controlled across stimulus sets.

Treatment procedures

Children were provided treatment three times weekly in individual sessions, each one hour in duration. Treatment followed a conventional protocol (Gierut, Reference Gierut, Dinnsen and Gierut2008a) consisting of two steps. The first was imitation of the treated sound in treated words. During imitation, a child viewed a picture of the treated word displayed on a computer monitor; the clinician said the depicted word; the child repeated that word. Each treated word was presented as a discrete trial, and feedback about accuracy of production was provided on a continuous schedule. Accuracy of production was assessed for the treated sound in the treated word-initial position; accurate production of the word form was not also required, in keeping with conventional principles of phonological treatment (Hodson & Paden, Reference Hodson and Paden1991: 119). In cases of inaccurate production of the treated sound, instructions about articulatory placement were provided, and an additional model was presented. Imitation continued until a child achieved 75% accuracy of production of the treated sound in the treated words over two consecutive sessions or until seven total sessions were completed, whichever occurred first. Following this, treatment shifted to spontaneous naming. The child again viewed the picture of the treated word on a computer monitor, but was to independently name that item with accurate production of the treated sound. Feedback and instruction were provided in the same manner outlined for imitation. This continued until a child achieved 90% accuracy of production over three consecutive sessions or until twelve total sessions were completed, whichever occurred first.

In all, children were afforded a maximum of nineteen treatment sessions (i.e. nineteen hours of training). They had the further opportunity to complete the protocol more rapidly, if they achieved performance levels of 75% and 90% accuracy of production over consecutive imitation and spontaneous sessions, respectively. Thus, time- and/or performance-based criteria determined time in treatment for a given child. In this study, those assigned to the dense conditions tended to follow time-based advancement through the protocol, whereas those assigned to the sparse conditions tended to follow performance-based advancement. Overall, the average number of sessions provided to any given child was fifteen, and the average by condition was nineteen for Dense–Freq, nineteen for Dense–Infreq, seven for Sparse–Freq and fourteen for Sparse–Infreq. Across conditions, there was no statistically significant difference in the number of sessions provided in each step of treatment (χ² (3, N=117)=1·12, p=0·77). Overall, mean accuracy of the treated sound at completion of treatment was 94%. Mean accuracy by condition was 93% for Dense–Freq, 85% for Dense–Infreq, 99% for Sparse–Freq and 96% for Sparse–Infreq. It must be underscored that performance during treatment showed only that children learned what was taught. The critical question was how children then transferred what had been learned to the expressive phonology. Treatment thus served as the conduit from which children were left free to generalize in keeping with the dependent variable.

Data analysis

Four sets of analyses of longitudinal generalization data were completed. Recall that, for each child, at each PKP sampling point in time, percentages of production accuracy were computed for each sound excluded from the phonemic inventory. This included the treated sound as sampled across contexts and words, as well as other untreated (erred) sounds. Any sound that evidenced any gain at any sampling point in time was entered into analysis, and the full set of longitudinal data associated with that sound was evaluated. Approximately 11,356 productions were examined for generalization, with each child contributing at least 1,420 productions over time.

The first two evaluations were complementary in establishing which experimental conditions induced generalization from qualitative and quantitative vantages, respectively. The analyses were identical to prior studies of density/frequency in treatment of PD (Morrisette & Gierut, Reference Morrisette and Gierut2002), and for comparison purposes, were again applied herein. Qualitatively, a dichotomous yes/no coding of generalization was applied to each experimental condition. The coding of yes/no generalization was based on the accepted criterion cut-off of 10% or greater mean generalization accuracy relative to baseline performance for children of a given condition (Elbert, Dinnsen & Powell, Reference Elbert, Dinnsen and Powell1984). This analysis offered preliminary insight to the experimental conditions affiliated with generalization. The quantitative analysis followed, using the rank procedure for replicated AB multiple baseline design to establish generalization as a statistically reliable effect (Morrisette & Gierut, Reference Morrisette and Gierut2002: 150). Briefly, the procedure makes use of the combined Sign test (Marascuilo & Serlin, Reference Marascuilo and Serlin1988), organizing sounds that generalized into three sets or ‘factors’ for each child by condition. The three factors were the treated sound, sounds from the same manner class as the treated (within-class generalization) and sounds from different manner classes than the treated (across-class generalization). The number of factors varied depending on which sounds generalized, e.g. if a given child showed no generalization to the treated sound, then that factor was not instantiated (see Table 2). Mean baseline performance and longitudinal generalization accuracy were computed and ranked for each factor, with the difference between ranks totaled by experimental condition. A binomial distribution, where p=0·05, was then applied to determine whether the resulting generalization was statistically reliable.

The third evaluation stands apart in its calculation of effect size for single-subject experimental design. This is a relatively new addition to the single-subject literature that has been introduced as a way of quantifying treatment efficacy, with an eye toward meta-analyses of small-n studies (Beeson & Robey, Reference Beeson and Robey2006). Effect size captures the magnitude of generalization for a given treatment condition, with values compared across conditions to identify which is relatively most efficacious. In this study, standard mean difference (d; Beeson & Robey, Reference Beeson and Robey2006) was computed. Mean accuracy of baseline data was calculated, along with mean accuracy of longitudinal generalization data. The difference between means was obtained and divided by the standard deviation of the baseline, corrected for continuity (Gierut & Morrisette, Reference Gierut and Morrisette2011b), to achieve the effect size, d. While benchmarks for interpretation of d have been established for between-group designs, it is inappropriate to extend these to small-n studies (Beeson & Robey, Reference Beeson and Robey2006). Standards for what constitutes a small, medium or large effect must be developed empirically, but before this can be established, reports of effect size in single-subject studies must accumulate. To our knowledge, there are only four studies enrolling clinical populations in some form of linguistic treatment that have cited effect sizes, with d values in the range of 2·0 to 18·0 (Beeson & Robey, Reference Beeson and Robey2006; Gierut & Morrisette, Reference Gierut and Morrisette2011a; Reference Gierut and Morrisette2011b). Data from the present study added to this base as a secondary contribution.

The three aforementioned analyses (i.e. criterion cut-off, rank procedure, effect size) were applied to the generalization data in comparisons blocked by density (dense vs. sparse neighborhoods) and by frequency (frequent vs. infrequent words), as well as in the orthogonal comparison of density×frequency. Together, the results revealed the triggers of generalization, namely, which type of treated words prompted generalization in the expressive phonology of children with PD.

For the fourth analysis, attention shifted from an evaluation of percentage accuracies to a descriptive characterization of the lexical properties of the PKP words that actually generalized. While certain words might trigger generalization, it was reasonable to also ask which words were the targets of generalization or the recipients of that change (Morrisette, Reference Morrisette2000). Consider, for example, that treatment in the Dense–Freq condition might only result in generalization to other frequent words in dense neighborhoods. To address this, the density and frequency values of PKP words that showed improvement in production accuracy from baseline to post-treatment were retrieved from the Hoosier Mental Lexicon (Nusbaum et al., Reference Nusbaum, Pisoni and Davis1984). Chi-square analyses were computed to determine whether there were differences in which words generalized based on neighborhood or frequency characteristics. The collective set of analyses thus identified both the triggers and targets of generalization.