- Email: [email protected]
Selection of predictor variables in assessing the severity of malocclusion
T. J. Freer, Brisbane,
B.D.Sc., Ph.D.,
Queensland,
F.D.S.R.C.S.,
D.Orth.*
Awtralia
I
n attempting to measure the over-all severity of a person’s malocclusion, several aspects of the problem are not always explicitly stated. Severity is generally accepted to be a continuous variable, ranging from ideal to very severe, although only a limited number of discrete points along the continuous scale are employed. However, very little is known about the processes by which a clinician arrives at an over-all severity score when it is demanded of him. The abstract nature of an over-all severity assessment is almost certainly a source of intraexaminer and interexaminer variation. It has been shown that this variability may be high.l A severity assessment may consist initially of a search for certain pathognomonic traits. Depending on the presence or absence of these traits, the clinician then looks at other secondary traits in arriving at a recommendation for or against treatment. It is also probable that previous clinical experience influences this decision. If, for example, the clinician can be reasonably sure that a certain condition will relapse if treated, he is more likely to recommend against treatment. Quantitative methods for estimating a person’s over-all severity score, usually implying an estimation of the need for orthodontic treatment, are dependent on the selection of a set of measured variables of the occlusion which may be used to determine that score. A large number of variables may be measured. Some method is required to determine which of these are used by the clinician in making a severity assessment. Observation of the different variables employed in previous indices of malocclusion strongly suggests that selection of the best set of predictors by means of clinical experience alone is not reliable. It is assumed that, a comprehensive set of variables will include the important *Dental
School,
University
of
Queensland,
Brisbane,
australia.
155
156
Table
Am. J. Orthod. August 1973
Freer I. Variables
used
in the
calculation
of
regression
equations
Abbreviation
Original Variables 1.
Overjet
2.
Labiolingual
OVJET spread ** hnaximum
value upper)
3.
Labiolingual
spread
(sum of upper
4.
Labiolingual
spread
(maximum
5.
Labiolingual
spread
(sum of lower
6.
Anterior
spacing
upper
spacing
lower
7.
Anterior Arch
length-upper
right
9.
Arch
length-upper
left minus
width
10.
Arch
Lingual
12.
Buccal
13.
Missing
14.
Upper
Composite
m
value lower)
LLSML
values)
LLSSL
ASL minus
minus
lower lower
right
ALURMLR ALULMLL
left
AWL3M4
m
crossbite-number
of posterior
crossbite-number
of posterior
teeth-number median
LLSSU
ASU
8.
11.
LLSMU
values)
of posterior
teeth
LXBT
teeth
BXBT MPTNO
teeth
diastema
UMD
Variables OVJET**2
1.
Overjet
squared
2.
Overjet
cubed
3.
Labiolingual
spread
(maximum
upper
value)
squared
LLSMU**2
4.
Labiolingual
spread
(maximum
upper
value)
cubed
LLSMU**3
5.
Cross-product (maximum
OVJET**3
of overjet upper
and labiolingual
spread
OVJ*LLS
value)
--
* XI
All distance Draker.
measurements
made to nearest
whole
millimetre.
1 9604.
predictors. For obvious reasons, however, the minimum number of predictors consistent with accuracy should be employed. Multiple regression offers a pragmatic and simple approach to the investigation of this problem, although th’e assumption of a linear model is not necessarily correct. A perplexing diEculty in applying multiple regression methods is the magnitude of the error of measurement in the criterion variable (severity). It has been shown not only that this error may be large but also that the error variance in the criterion is dependent on the magnitude of the criterion. As the severity of malocclusion increases, error variance decreases1 The methods which may be used to deal with the problem of unequal error variance, such as transformations, will not be discussed further here. In spite of any shortcomings, multiple regression techniques are useful in the preliminary investigation of a set of measured variables in order to determine which variables influence the clinician’s assessment of severity and are therefore most likely to be useful in
Volume Number
Table
64 2
Xelection of predictor
II. Stepwise
Step
multipie
Variable Entered
regression
OF severity
R
score
(O-10)
K2
on twelve
variables predictor
I
variables
increase in R*
ResiduaE Mean Square
OVJET
0.622
0.387
0.387
3.355
2
LLSMU
0.764
0.584
0.207
0.933
3
BXBT
0.780
0.608
0.024
0.891
4
AWL3M4
0.791
0.626
0.018
0.864
5
ASU
0.797
0.635
0.009
0.856
6
0.643
0.008
0.849
ASL
0.802
7
MPTNO
0.807
0.651
0.008
0.842
8
UMD
0.809
0.654
0.003
0.850
9
LLSSU
0.811
0.658
0.004
0.852
10
LLSSL
0.813
0.661
0.003
0.861
11
LXBT
0.815
0.664
0.003
0.867
12
LLSML
0.815
0.665
0.001
0.878
any quantitative assessment system. The present report concerns such an investigation, and its aim is purely expository. It is not designed to provide an index of the severity of malocclusion. Materials
and
methods
A sample of seventy-two sets of study models taken in upstate New York and Iowa was used. The present study was confined to persons with distocclusion as defined by Grainger. 2 Fourteen predictor variables were measured on each set of study models. In addition, five new predictor variables mere derived from two of the original fourteen predictor variables. These five new L’composite” variables were the second-order (squared) and third-order (cubed) terms of the original measurement of overjet, the second- and third-order terms of the original measurement of labiolingual spread (maximum upper value), and the cross-product term of overjet and labiolingual spread (maximum upper value). All predictor variables are shown in Table I. The dependent or criterion variable used in all analyses was the 0 to POseverity score of Grainger.” Each set of study models was evaluated by three orthodontists, and a score on the 0 to 10 scale was given, The mean of the three scores, to the nearest whole number, was used as the criterion variable. Stepwise multiple linear regressions were performed on the data, using the BMD” and SPSSt statistical packages. Standard methods were used to evaluate the regression analyses, including plots of normalized residuals and plots of residuals against the criterion and predictor variables. “Dixon, W. J. (Editor) : Biomedical California Press. SNie, N., Bent, D. H., and Hull, Company, Inc.
Computer C.
H.
Programs,
(Editors)
: New
Los York,
Angeles, 1970,
1970,
University
M&raw-Hill
of Book
158
Am. J. Orthod. August 1973
Freer
l’able III. Stepwise multiple including --
higher
order
regression severity score (O-10) on thirteen predictor and cross-product terms of the original variables
Variable Entered
1
OVJ’LLS
0.668
0.446
0.446
1.224
2
OVJET**2
0.716
0.513
0.067
1.091
I32
Increase in R2
Residual Mean Square
step
R
variables,
3
LLSMU
0.812
0.660
0.147
0.774
4
LLSML
0.827
0.684
0.024
0.730
5
MPTNO
0.836
0.700
0.016
0.704
6
LLSMU**2
0.843
0.710
0.010
0.689
7
BXBT
0.850
0.723
0.012
0.670
8
OVJET
0.864
0.729
0.006
0.666
9
LLSSU
0.857
0.734
0.005
0.664
10
LLSMU*“3
0.861
0.741
0.007
0.658
11
ASU
0.862
0.744
0.003
0.660
12
ASL
0.864
0.747
0.003
0.662
13
AWL3M4
0.866
0.749
0.002
0.668
Results
A preliminary regression on the original fourteen predictor variables showed that two variables added virtually nothing to the fit of the f%ial equation. These two variables (arch length-upper right minus lower right ; arch lengthupper left minus lower left) also contained missing data values. Because the residual plotting facilities of the program BMDOBR were to be employed and this program would not delete missing data, the two variables containing missing values were not considered further. The stepwise regression on the remaining twelve variables is summarized in Table II. A final multiple correlation of 0.82 was obtained ; however, the residual mmean square value began to rise after Step 8. The increase in the percentage of explained variance (R2) after Step 2 was, small, and after Step 4 it was less than 1 per cent at each step. The 65 per cent of explained variance at Step 8 has risen only 3 per cent from Step 4. A plot of normalized residuals from this regression showed that the fit of the model for negative residuals was poor and that second- and third-order terms fo-r some of the predictors might be more appropriate. Previous plots of the criterion against two of the predictors also suggested the presence of a seeondor third-order relationship. Accordingly, second- and third-order terms for the first two variables were investigated. In addition, the cross-product term of these two variables was investigated. Therefore, five more composite predictor variables were added to the original twelve variables, making a total of seventeen variables. A stepwise regression of severity on these seventeen variables was then carried out, giving a final multiple correlation coefficient of 0.866. Evaluation of the residual mean square values and the increase in the percentage of ex-
Selection
Volume Nunzbei-
64 2
Table
IV.
Stepwise
multiple
regression
of
severity
score
of predictor (O-10)
an
vartibles thirteen
1
predictor
variables Step
Variable Entered
I32
R
Increase in R2
Yean Square
1
DUMOOl NM002
0.195
0.038
0.038
2.156
2
OVJ*LLS
0.691
0.478
0.440
1.187
3
DVJET**2
0.740
0.547
0.069
1.045
4
LLSMU
0.830
0.688
0.141
0,730
5
LLSML
0.848
8.718
0.030
0.670
6
OVJET
0.859
0.739
0.021
0.632
7
LLSSU
0.868
0.753
0.014
0.607
8
MPTMO
0.873
0.763
0.010
0.592
9
ASU
0.876
0.768
0.005
0.589
10
LLSMU**2
0.878
0.772
0.004
0.588
11
LLSMU**3
0.885
0.782
0.011
0.570
12
ASL
0.886
0.786
0.004
0.571
13
LAW3M4
0.887
0.787
0.001
0.579
*One of these variables present = 1).
(buccal cross-bite)
was used as a dummy variable1 (a.bsent = 0 3
plained variance due to regression at each step suggested that at least four of these seventeen variables should be discarded. The four variables discarded were the values for upper median diastema, lingual cross-bite, third-order (cubed) term for overjet, and labiolingual spread (sum of lower values). The stepwise regression on the remaining thirteen variables is shown in Table III. This gave a final multiple correlation coefficient of 0.866, the same as for the regression on all seventeen variables. It should be noted that the residual mean square value begins to rise after Step 10. This suggests that even more of these variables could be discarded. Inclusion of the new composite terms has given an improvement over the regression on twelve variables (Table II). Of particular interest is the importance of the cross-product term and the second-order term for overjet. Previous plots of the criterion (severity) and residuals against raw scores of buccal cross-bite indicated that this variable was more important when considered on a presence-absence basis rather than as a continuous variable (number of teeth in buccal cross-bite) ~ -4 further regression was carried out using buceal cross-bite as a dummy variable (absent = 0, present = 1) which was forced in on the first step (Table IV). A further improvement in the result was seen and the residual mean square value began to rise after Step 11, the value of R at this step being 0.885. Discussion
The quantitative estimation of the over-all severity of malocclusion requires, for reasons of efficiency and cost, the minimum number of predictors consistent
160
Freer
Am. J. Orthod. August 1973
with accuracy. It is desirable, therefore, not only to eliminate those variables which do not influence the clinician’s assessment but also to reduce the numbers of highly intercorrelated variables, which are equally good predictors. Multiple stepwise regression is a reasonably simple method for achieving such reduction. The danger is that some variables will be overlooked because of the particular composition of the sample used. The other technique for identifying important variables, which is now widely available in statistical computer packages, is discriminant function analysis. This, however, suffers from the disadvantage of requiring the prior identification of groups, which, at present, cannot be done confidently. In the initial stages, however, these two techniques may be complementary, particularly with the rigorous analysis of residuals in regression in I order to identify outliers. One of the predictor variables used in the present study has not previously been used extensively-labiolingual spread.4 Although it reflects anterior crowding, it is more highly correlated with severity than other measures of crowding, such as metric assessments of space shortage. Labiolingual spread values will be increased with relative displacement of anterior teeth from any cause such as ectopic eruption. It may be speculated, therefore, that the clinician is not so much interested in crowding per se as in the displacement that it causes. Of the twelve original variables, the highest correlation for the severity score was with overjet (0.62) and the next highest was with labiolingual spreadmaximum upper value (0.55). The correlation of severity with the cross-product term of these two variables (0.67) was higher than for either variable alone. Also of interest were the correlations of severity with the four variables dependent on this same feature, that is, labiolingual spread. Severity was more highly correlated with the maximum upper value for labiolingual spread (0.55) than with the sum of the upper values (0.49). The correlations with the equivalent lower arch variables were much lower (0.20 and 0.21). The present sample is composed of distocclusion cases only, but earlier studies3 indicate that overjet is an important predictor of severity in all types of malocclusion. This point draws attention to the difficulty of interpreting estimating equations for samples of mixed buccal segment classes. Almost certainly, clinicians give different weights to various predictors, depending on the buccal segment classification. Again, experience has shown that estimations of severity, regardless of technique, will be more accurate if different buccal segment classes are separated first. It may be that Angle’s classification is so ingrained in orthodontic diagnosis that it influences the clinician unduly. Alternatively, Angle’s classification may be based on just that variable which clinicians tend to assessfirst before making an o-ver-all assessment. The importance of the cross-product term in the present analyses suggests that the clinician assesses at least some variables in the light of the values of other variables-that is, there is a strong interactive effect. While the researcher is usually limited in the number of regression analyses that may be made for any one set of data, more attention should be given to the influence of combinations of variables on the clinician. The introduction of buccal cross-bite as a
Volume Number
64 2
Selection
of
predictor
variables
1
dummy variable and t,he resulting improvement in the regression solution support the idea that the clinician does some prior sorting of variables before making an assessment. Summary
Multiple regression techniques have been used to investigate the importance to the clinician of different variables when making an over-a,11 assessment of the severity of malocclusion. The results indicated that most of the variance in the criterion (severity) was explained by a limited number of predictors. One of the predictors which has not been widely used (labiolingual spread) was shown to be very important. There was shown to be an important interactive effect of at least two of the predictors (overjet and labiolingual spread). When one of the predictors (buccal cross-bite) was used as a dummy variable, the regression fit was further improved. In future investigations, attention should be given to the effects of combinations of variables on the clinician’s assessments of severity. and
I would like Dr. Naham
to express Cons for
my their
thanks to Dr. James help with all phases
P. Carlos, Dr. J. Grewe, of this project.
Mr.
R’. Senning,
REFERENCES
1. Freer, T. J., Grewe, J. M., and Little, R. M.: Agreement among the subjective severity asses’sments of 10 orthodontists, Angle Orthod. 43: 42-45, 1973. 2. Grainger, R. M.: Orthodontic treatment priority index, Ser. 2, No. 25, Washington, 1967, United States Department of Health, Education and Welfare. 3. Freer, T. J.: Assessment The matched pair similarity technic, Int. of occlusal status: Dent. J. 22: 412-422, 1972. 4. Draker, H. L. : Handicapping labiolingual deviations : A proposed index for public health purposes, AM. J. ORTHOD. 46: 295-305, 1960. Twrbot
St.
B.D.Sc., Ph.D.,
Queensland,
F.D.S.R.C.S.,
D.Orth.*
Awtralia
I
n attempting to measure the over-all severity of a person’s malocclusion, several aspects of the problem are not always explicitly stated. Severity is generally accepted to be a continuous variable, ranging from ideal to very severe, although only a limited number of discrete points along the continuous scale are employed. However, very little is known about the processes by which a clinician arrives at an over-all severity score when it is demanded of him. The abstract nature of an over-all severity assessment is almost certainly a source of intraexaminer and interexaminer variation. It has been shown that this variability may be high.l A severity assessment may consist initially of a search for certain pathognomonic traits. Depending on the presence or absence of these traits, the clinician then looks at other secondary traits in arriving at a recommendation for or against treatment. It is also probable that previous clinical experience influences this decision. If, for example, the clinician can be reasonably sure that a certain condition will relapse if treated, he is more likely to recommend against treatment. Quantitative methods for estimating a person’s over-all severity score, usually implying an estimation of the need for orthodontic treatment, are dependent on the selection of a set of measured variables of the occlusion which may be used to determine that score. A large number of variables may be measured. Some method is required to determine which of these are used by the clinician in making a severity assessment. Observation of the different variables employed in previous indices of malocclusion strongly suggests that selection of the best set of predictors by means of clinical experience alone is not reliable. It is assumed that, a comprehensive set of variables will include the important *Dental
School,
University
of
Queensland,
Brisbane,
australia.
155
156
Table
Am. J. Orthod. August 1973
Freer I. Variables
used
in the
calculation
of
regression
equations
Abbreviation
Original Variables 1.
Overjet
2.
Labiolingual
OVJET spread ** hnaximum
value upper)
3.
Labiolingual
spread
(sum of upper
4.
Labiolingual
spread
(maximum
5.
Labiolingual
spread
(sum of lower
6.
Anterior
spacing
upper
spacing
lower
7.
Anterior Arch
length-upper
right
9.
Arch
length-upper
left minus
width
10.
Arch
Lingual
12.
Buccal
13.
Missing
14.
Upper
Composite
m
value lower)
LLSML
values)
LLSSL
ASL minus
minus
lower lower
right
ALURMLR ALULMLL
left
AWL3M4
m
crossbite-number
of posterior
crossbite-number
of posterior
teeth-number median
LLSSU
ASU
8.
11.
LLSMU
values)
of posterior
teeth
LXBT
teeth
BXBT MPTNO
teeth
diastema
UMD
Variables OVJET**2
1.
Overjet
squared
2.
Overjet
cubed
3.
Labiolingual
spread
(maximum
upper
value)
squared
LLSMU**2
4.
Labiolingual
spread
(maximum
upper
value)
cubed
LLSMU**3
5.
Cross-product (maximum
OVJET**3
of overjet upper
and labiolingual
spread
OVJ*LLS
value)
--
* XI
All distance Draker.
measurements
made to nearest
whole
millimetre.
1 9604.
predictors. For obvious reasons, however, the minimum number of predictors consistent with accuracy should be employed. Multiple regression offers a pragmatic and simple approach to the investigation of this problem, although th’e assumption of a linear model is not necessarily correct. A perplexing diEculty in applying multiple regression methods is the magnitude of the error of measurement in the criterion variable (severity). It has been shown not only that this error may be large but also that the error variance in the criterion is dependent on the magnitude of the criterion. As the severity of malocclusion increases, error variance decreases1 The methods which may be used to deal with the problem of unequal error variance, such as transformations, will not be discussed further here. In spite of any shortcomings, multiple regression techniques are useful in the preliminary investigation of a set of measured variables in order to determine which variables influence the clinician’s assessment of severity and are therefore most likely to be useful in
Volume Number
Table
64 2
Xelection of predictor
II. Stepwise
Step
multipie
Variable Entered
regression
OF severity
R
score
(O-10)
K2
on twelve
variables predictor
I
variables
increase in R*
ResiduaE Mean Square
OVJET
0.622
0.387
0.387
3.355
2
LLSMU
0.764
0.584
0.207
0.933
3
BXBT
0.780
0.608
0.024
0.891
4
AWL3M4
0.791
0.626
0.018
0.864
5
ASU
0.797
0.635
0.009
0.856
6
0.643
0.008
0.849
ASL
0.802
7
MPTNO
0.807
0.651
0.008
0.842
8
UMD
0.809
0.654
0.003
0.850
9
LLSSU
0.811
0.658
0.004
0.852
10
LLSSL
0.813
0.661
0.003
0.861
11
LXBT
0.815
0.664
0.003
0.867
12
LLSML
0.815
0.665
0.001
0.878
any quantitative assessment system. The present report concerns such an investigation, and its aim is purely expository. It is not designed to provide an index of the severity of malocclusion. Materials
and
methods
A sample of seventy-two sets of study models taken in upstate New York and Iowa was used. The present study was confined to persons with distocclusion as defined by Grainger. 2 Fourteen predictor variables were measured on each set of study models. In addition, five new predictor variables mere derived from two of the original fourteen predictor variables. These five new L’composite” variables were the second-order (squared) and third-order (cubed) terms of the original measurement of overjet, the second- and third-order terms of the original measurement of labiolingual spread (maximum upper value), and the cross-product term of overjet and labiolingual spread (maximum upper value). All predictor variables are shown in Table I. The dependent or criterion variable used in all analyses was the 0 to POseverity score of Grainger.” Each set of study models was evaluated by three orthodontists, and a score on the 0 to 10 scale was given, The mean of the three scores, to the nearest whole number, was used as the criterion variable. Stepwise multiple linear regressions were performed on the data, using the BMD” and SPSSt statistical packages. Standard methods were used to evaluate the regression analyses, including plots of normalized residuals and plots of residuals against the criterion and predictor variables. “Dixon, W. J. (Editor) : Biomedical California Press. SNie, N., Bent, D. H., and Hull, Company, Inc.
Computer C.
H.
Programs,
(Editors)
: New
Los York,
Angeles, 1970,
1970,
University
M&raw-Hill
of Book
158
Am. J. Orthod. August 1973
Freer
l’able III. Stepwise multiple including --
higher
order
regression severity score (O-10) on thirteen predictor and cross-product terms of the original variables
Variable Entered
1
OVJ’LLS
0.668
0.446
0.446
1.224
2
OVJET**2
0.716
0.513
0.067
1.091
I32
Increase in R2
Residual Mean Square
step
R
variables,
3
LLSMU
0.812
0.660
0.147
0.774
4
LLSML
0.827
0.684
0.024
0.730
5
MPTNO
0.836
0.700
0.016
0.704
6
LLSMU**2
0.843
0.710
0.010
0.689
7
BXBT
0.850
0.723
0.012
0.670
8
OVJET
0.864
0.729
0.006
0.666
9
LLSSU
0.857
0.734
0.005
0.664
10
LLSMU*“3
0.861
0.741
0.007
0.658
11
ASU
0.862
0.744
0.003
0.660
12
ASL
0.864
0.747
0.003
0.662
13
AWL3M4
0.866
0.749
0.002
0.668
Results
A preliminary regression on the original fourteen predictor variables showed that two variables added virtually nothing to the fit of the f%ial equation. These two variables (arch length-upper right minus lower right ; arch lengthupper left minus lower left) also contained missing data values. Because the residual plotting facilities of the program BMDOBR were to be employed and this program would not delete missing data, the two variables containing missing values were not considered further. The stepwise regression on the remaining twelve variables is summarized in Table II. A final multiple correlation of 0.82 was obtained ; however, the residual mmean square value began to rise after Step 8. The increase in the percentage of explained variance (R2) after Step 2 was, small, and after Step 4 it was less than 1 per cent at each step. The 65 per cent of explained variance at Step 8 has risen only 3 per cent from Step 4. A plot of normalized residuals from this regression showed that the fit of the model for negative residuals was poor and that second- and third-order terms fo-r some of the predictors might be more appropriate. Previous plots of the criterion against two of the predictors also suggested the presence of a seeondor third-order relationship. Accordingly, second- and third-order terms for the first two variables were investigated. In addition, the cross-product term of these two variables was investigated. Therefore, five more composite predictor variables were added to the original twelve variables, making a total of seventeen variables. A stepwise regression of severity on these seventeen variables was then carried out, giving a final multiple correlation coefficient of 0.866. Evaluation of the residual mean square values and the increase in the percentage of ex-
Selection
Volume Nunzbei-
64 2
Table
IV.
Stepwise
multiple
regression
of
severity
score
of predictor (O-10)
an
vartibles thirteen
1
predictor
variables Step
Variable Entered
I32
R
Increase in R2
Yean Square
1
DUMOOl NM002
0.195
0.038
0.038
2.156
2
OVJ*LLS
0.691
0.478
0.440
1.187
3
DVJET**2
0.740
0.547
0.069
1.045
4
LLSMU
0.830
0.688
0.141
0,730
5
LLSML
0.848
8.718
0.030
0.670
6
OVJET
0.859
0.739
0.021
0.632
7
LLSSU
0.868
0.753
0.014
0.607
8
MPTMO
0.873
0.763
0.010
0.592
9
ASU
0.876
0.768
0.005
0.589
10
LLSMU**2
0.878
0.772
0.004
0.588
11
LLSMU**3
0.885
0.782
0.011
0.570
12
ASL
0.886
0.786
0.004
0.571
13
LAW3M4
0.887
0.787
0.001
0.579
*One of these variables present = 1).
(buccal cross-bite)
was used as a dummy variable1 (a.bsent = 0 3
plained variance due to regression at each step suggested that at least four of these seventeen variables should be discarded. The four variables discarded were the values for upper median diastema, lingual cross-bite, third-order (cubed) term for overjet, and labiolingual spread (sum of lower values). The stepwise regression on the remaining thirteen variables is shown in Table III. This gave a final multiple correlation coefficient of 0.866, the same as for the regression on all seventeen variables. It should be noted that the residual mean square value begins to rise after Step 10. This suggests that even more of these variables could be discarded. Inclusion of the new composite terms has given an improvement over the regression on twelve variables (Table II). Of particular interest is the importance of the cross-product term and the second-order term for overjet. Previous plots of the criterion (severity) and residuals against raw scores of buccal cross-bite indicated that this variable was more important when considered on a presence-absence basis rather than as a continuous variable (number of teeth in buccal cross-bite) ~ -4 further regression was carried out using buceal cross-bite as a dummy variable (absent = 0, present = 1) which was forced in on the first step (Table IV). A further improvement in the result was seen and the residual mean square value began to rise after Step 11, the value of R at this step being 0.885. Discussion
The quantitative estimation of the over-all severity of malocclusion requires, for reasons of efficiency and cost, the minimum number of predictors consistent
160
Freer
Am. J. Orthod. August 1973
with accuracy. It is desirable, therefore, not only to eliminate those variables which do not influence the clinician’s assessment but also to reduce the numbers of highly intercorrelated variables, which are equally good predictors. Multiple stepwise regression is a reasonably simple method for achieving such reduction. The danger is that some variables will be overlooked because of the particular composition of the sample used. The other technique for identifying important variables, which is now widely available in statistical computer packages, is discriminant function analysis. This, however, suffers from the disadvantage of requiring the prior identification of groups, which, at present, cannot be done confidently. In the initial stages, however, these two techniques may be complementary, particularly with the rigorous analysis of residuals in regression in I order to identify outliers. One of the predictor variables used in the present study has not previously been used extensively-labiolingual spread.4 Although it reflects anterior crowding, it is more highly correlated with severity than other measures of crowding, such as metric assessments of space shortage. Labiolingual spread values will be increased with relative displacement of anterior teeth from any cause such as ectopic eruption. It may be speculated, therefore, that the clinician is not so much interested in crowding per se as in the displacement that it causes. Of the twelve original variables, the highest correlation for the severity score was with overjet (0.62) and the next highest was with labiolingual spreadmaximum upper value (0.55). The correlation of severity with the cross-product term of these two variables (0.67) was higher than for either variable alone. Also of interest were the correlations of severity with the four variables dependent on this same feature, that is, labiolingual spread. Severity was more highly correlated with the maximum upper value for labiolingual spread (0.55) than with the sum of the upper values (0.49). The correlations with the equivalent lower arch variables were much lower (0.20 and 0.21). The present sample is composed of distocclusion cases only, but earlier studies3 indicate that overjet is an important predictor of severity in all types of malocclusion. This point draws attention to the difficulty of interpreting estimating equations for samples of mixed buccal segment classes. Almost certainly, clinicians give different weights to various predictors, depending on the buccal segment classification. Again, experience has shown that estimations of severity, regardless of technique, will be more accurate if different buccal segment classes are separated first. It may be that Angle’s classification is so ingrained in orthodontic diagnosis that it influences the clinician unduly. Alternatively, Angle’s classification may be based on just that variable which clinicians tend to assessfirst before making an o-ver-all assessment. The importance of the cross-product term in the present analyses suggests that the clinician assesses at least some variables in the light of the values of other variables-that is, there is a strong interactive effect. While the researcher is usually limited in the number of regression analyses that may be made for any one set of data, more attention should be given to the influence of combinations of variables on the clinician. The introduction of buccal cross-bite as a
Volume Number
64 2
Selection
of
predictor
variables
1
dummy variable and t,he resulting improvement in the regression solution support the idea that the clinician does some prior sorting of variables before making an assessment. Summary
Multiple regression techniques have been used to investigate the importance to the clinician of different variables when making an over-a,11 assessment of the severity of malocclusion. The results indicated that most of the variance in the criterion (severity) was explained by a limited number of predictors. One of the predictors which has not been widely used (labiolingual spread) was shown to be very important. There was shown to be an important interactive effect of at least two of the predictors (overjet and labiolingual spread). When one of the predictors (buccal cross-bite) was used as a dummy variable, the regression fit was further improved. In future investigations, attention should be given to the effects of combinations of variables on the clinician’s assessments of severity. and
I would like Dr. Naham
to express Cons for
my their
thanks to Dr. James help with all phases
P. Carlos, Dr. J. Grewe, of this project.
Mr.
R’. Senning,
REFERENCES
1. Freer, T. J., Grewe, J. M., and Little, R. M.: Agreement among the subjective severity asses’sments of 10 orthodontists, Angle Orthod. 43: 42-45, 1973. 2. Grainger, R. M.: Orthodontic treatment priority index, Ser. 2, No. 25, Washington, 1967, United States Department of Health, Education and Welfare. 3. Freer, T. J.: Assessment The matched pair similarity technic, Int. of occlusal status: Dent. J. 22: 412-422, 1972. 4. Draker, H. L. : Handicapping labiolingual deviations : A proposed index for public health purposes, AM. J. ORTHOD. 46: 295-305, 1960. Twrbot
St.