pn=2
pn=1
pn=2-pn=1
Figure 2. The weight, p, of the proportion of consonants correct (PCC) versus the proportion of consonants in targeted speech, PC, for two different values of the relative weight, n, between correctly produced consonants and all phones. The difference in the weight p versus PC is also shown.
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PC values for typical words in speech samples exceed 55% for stress-timed languages, like English, and 50% for syllabic languages, like Greek and Spanish. Typical monosyllabic word samples that are used by one of the authors (D. I.) to differentiate normal from disordered child speech in English have the following PC values: 64% for monosyllabic without consonant clusters, 76% for monosyllabic words with a consonant cluster. The corresponding p values computed from (3) with n=1 are respectively 0.39 and 0.43. Therefore, if PWP for the consonant cluster words is computed using the p corresponding to the words with only singleton consonants, it will only differ from the true PWP by less than 4%. If n=2 instead of n=1 is used in computing p, then its values for the words with only consonant singletons and the words with a consonant cluster are respectively 0.56 and 0.60, resulting again in a very small error for PWP when using the p of the other word category.
Another example is given here using the data obtained by one of the authors (E. B.) from a child’s English speech at the age of 3 years. The child’s monosyllabic words with only singleton consonants have a PC equal to 59% while the monosyllabic words with a consonant cluster have a PC equal to 71%. The corresponding p values for n=1 are respectively 0.37 and 0.415. For n=2, they are 0.541 and 0.587. Again, the PWP computed using the p of the other word category would differ by an amount from the true PWP that can be neglected. Therefore, for most practical purposes, the conclusions drawn in the first case above, where p was invariant between speech samples, hold true here as well and they will not be repeated.
Third case
In the third case, the change in p will not be ignored across word categories. Ingram (2015) notes that there are cases of children’s disordered speech where PCC changes across words with clusters and words without clusters are negligible. For such cases, it will be useful to use such n as to increase the PWP change across the word categories. This PWP change is compared for two arbitrarily chosen values of n. Without going through the algebraic details, use of equation (3) four times, twice for each n to compute the PWP for each word category, results in
ΔPWP1 - ΔPWP2 = - (p2-p1) ΔPPD (10)
where Δ is the change of the quantity of interest (PWP or PPD) across word categories and the subscript refers to the first or second n used in computing p and PWP. In (10), p may be computed for either category as it will yield the same result. This is because the difference (p2-p1) changes negligibly for any changes in PC values larger than about 50%. This may be observed in Figure 2 where p1 (n=1) and p2 (n=2) and their difference is plotted for all possible PC values. Comparing (p2- p1) values at different PC values larger than about 50%, it is seen that they are practically the same.
For example, for PC=50%, p1=1/3 and p2=0.5 and, thus, p2-p1= 0.167. For PC=75%, p1= 0.429 and p2=0.6 and, thus, p2-p1= 0.171. The change in the difference (p2-p1) is indeed negligible. Derivation of equation (10) is based on this observation.
What does equation (10) imply for practical applications? Without loss of generality, let the subscript 1 refer to the smaller of the two n values chosen. Then p2-p1 is positive and for negative ΔPPD (for example PPD for monosyllabic words with only singleton consonants minus PPD for words with a consonant cluster), the left hand side becomes positive. For negative ΔPPD, ΔPWP is positive independent of the n chosen as ΔPCC=0, giving that ΔPWP is larger for the smaller n. Distinguishing PWP between categories of word complexity is sought in practice and, therefore, it is better in such cases, as the one considered here, to use as small an n as possible. Ingram’s proposition of n=1 is the smallest integer n that can used for optimal results. Furthermore, equation (10) gives the difference in the change of PWP for two arbitrary values of n, for a given ΔPPD. To get a feeling on the amount that this difference changes for different values of n, it is now computed for PC=60% and n=0.5, n=1, and n=2. ΔPWP for n=0.5 is larger than ΔPWP for n=1 by the amount 0.14 times the absolute value of ΔPPD. In turn, ΔPWP for n=1 is larger than ΔPWP for n=2 by the amount 0.17 times the absolute value of ΔPPD.
Proceedings ISMBS 2015
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Conclusion
Obtaining a formula for phonological word proximity (PWP) for a whole speech sample in terms of the proportion of consonants correct and the proportion of phonemes deleted, made it possible to examine the effect of the relative weight of phones and of the proportion of consonants in the phonemes on: a) the weight of each PWP component individually and in relation to each other, and b) the computed PWP and its sensitivity to measurements across different speech samples, in general, and across categories of word complexity in disordered child speech, in particular. The analysis and formulae given here provide guidelines to practitioners for child speech assessment. It is pointed out, however, that the present work applies mostly to normal or disordered child speech since targeted vowels are considered to be produced correctly when they are produced in context. The present work is being extended to also differentiate correct from incorrect vowels when they are produced in context. This will find applications in assessing second language speech where vowel mispronunciation occurs even in L2 learners at advanced levels.
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