# Logistic Regression and Linear Discriminant Analyses in Evaluating Factors Associated with Asthma Prevalence among 10- to 12-Years-Old Children: Divergence and Similarity of the Two Statistical Methods

Authors: George Antonogeorgos; Demosthenes B. Panagiotakos; Kostas N. Priftis; Anastasia Tzonou
Journal: International Journal of Pediatrics
 (2009)

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## Abstract

Logistic regression and discriminant analyses are both applied in order to predict the probability of a specific categorical outcome based upon several explanatory variables (predictors). The aim of this work is to evaluate the convergence of these two methods when they are applied in data from the health sciences. For this purpose, we modeled the association of several factors with the prevalence of asthma symptoms with both the two methods and compared the result. In conclusion, logistic and discriminant analyses resulted in similar models.


---

## Body

## 1. Introduction

Logistic regression and linear discriminant analyses
are multivariate statistical methods which can be used for the evaluation of
the associations between various covariates and a categorical outcome. Both
methodologies have been extensively applied in research, especially in medical
and sociological sciences. Logistic regression is a form of regression which is
used when the dependent variable is dichotomous, discrete, or categorical, and
the explanatory variables are of any kind. In medical sciences, the outcome is usually
the presence or absence of a stated situation or a disease. Using the logit
transformation, logistic regression predicts always the probability of group
membership in relation to several variables independent of their distribution. The
logistic regression analysis is based on calculating the odds of the outcome as
the ratio of the probability of having the outcome divided by the probability
of not having it. Discriminant analysis is a similar classification method that is used to
determine which set of variables discriminate between two or more naturally
occurring groups and to classify an observation into these known groups. In
order to achieve that discriminant analysis is based on the estimation of the
orthogonal discriminant functions, the linear combination of the standardized
independent predictor variables
gives the greatest means differences between the existing
groups. Thus, it can be proposed that both discriminant analysis and logistic
regression can be used to predict the probability of a specified outcome using
all or a subset of available variables.Although the theoretical properties
have been studied extensively throughout the literature, the choice of the
proper method in data analysis is still a question for the researcher. The aim
of this work after summarizing the properties of the two discriminating methods
is to explore the convergence
of the two analytical methods when they are used to evaluate categorical health
outcomes in the pediatric epidemiological research. In particular, we tested
the associations between anthropometric and lifestyle patterns in relation to
asthma prevalence among 10–12-year-old
children, using both statistical methods. So, the reader will elucidate the differences
and the similarities of the two methods in order to make the appropriate choice
in their application.


## 2. Material and Methods

### 2.1. Linear Discriminant Analysis

Discriminant analysis focuses on
the association between multiple independent variables and a categorical
dependent variable by forming a composite of the independent variables. This type of multivariate analysis
can determine the extent
of any of the composite variables discriminates between two or more pre-existing groups of
subjects and also can derive a classification model for predicting the group
membership of new observations [1]. The simplest type of discriminant analysis
is when the dependent variable has two groups. In this case, a linear
discriminant function that passes through the means of the two groups (centroids)
can be used to discriminate subjects between the two groups. When there are
more groups, the number of groups minus one function is needed to classify an
observation among them. For each of the groups, linear discriminant analysis assumes the explanatory
variables to be normally distributed with equal covariance matrices. For each case, the estimated
coefficient for an independent variable is multiplied by the case's score on
that variable. These products are summed and added to the constant, and the
result is a composite score, that is, the discriminant score for that case.The
linear discriminant function (LDF) is represented by(1)LDF=b0+b1xi1+b2xi2+⋯+bkxik=bX,   where bj is the
value of the jth coefficient, j=1,…,k, and xij is the value of the ith case of the jth predictor. The
LDF can also be written in standardized form which allows comparing variables
measured on different scales. In the standardized LDF, each variable is
adjusted by subtraction of its mean value and division by its standard deviation.
Coefficients with large absolute values
reflect greater discriminating ability to their corresponding variables. From the LDF, scores can
estimate predicted probabilities and predicted group membership for every case on
the dependent variable. This approach is based on the rationale that it is more
likely that the independent and dependent variables are related as the between-groups sum of square is larger relative to within-group sum of squares. Also the ratio of between-group divided by total sum
of squares (eta-squared statistic or explained variability) or of within-group
divided by total sum of squares (Wilks' lambda statistic or unexplained variability) is used to
assess the relationship. As we can see, the ratio of between-group divided by
within-group sum of squares is an analogue to the ratio of variances, which is
the F statistic, a test that controls the possibility that the observed
relationship is due to chance.The principle by which
the discriminant coefficients (or weights) are selected is that they maximize the
distance between the two group means (centroids)|y¯1−y¯2|.
Fisher [2] was the first who suggested to transform the multivariate
observation x to univariate observations y in such way that the y's
derived from groups 1 and 2 have the maximum distance between them. Thus, the
linear combination y=a′x is the one that maximizes the ratio (squared
distance between sample means)/(sample variance y). The vector of
coefficients is given by the eigenvectors of the matrix B*S−1, where B=(x¯1−x¯2)′ is the between-group matrix and S is an estimate of Σ. A very important characteristic
of these composite sums of squares is that they enclose the variability and the
covariability of each variable. The discriminant coefficients can be calculated
in unstandardized or standardized form but they are irrelatively of the form, less informative than
those in regression. Assuming
that there are 2 groups, x¯1,x¯2 are the means of each group, and S is the pooled covariance
matrix, the allocation rule based on Fisher's discriminant functions is the
following: (2)Xi∈{group1,ify=(x¯1−x¯2)′S−1Xi≥12(x¯1−x¯2)′S−1(x¯1+x¯2),group2,ify=(x¯1−x¯2)′S−1Xi<12(x¯1−x¯2)′S−1(x¯1+x¯2).


### 2.2. Logistic Regression Analysis

Logistic regression is a form of
regression which is used when we want to predict probabilities of the presence
or absence of a particular disease, characteristic, or an outcome in general
based on a set of independent of explanatory variables of any kind (continuous,
discrete, or categorical) [3]. Since the predicted probability must lie between 0 and 1, simple
linear regression techniques are insufficient to achieve that, because they
allow the dependent variable to pass these limits and to produce inconsistent
results.  Defined as P1, the probability of an object is belonging to group 1, and as P0, the probability of an object is belonging to group 0. The logistic
regression model has the form of (3)zi=log(Pi1Pi0)=b0+b1xi1+b2xi2+⋯+bkxik,  where Pi1/Pi0 is called the odds ratio, bj is the value of the jth coefficient, j=1,…,k, and xij is the value of the ith case of the jth predictor.
The parameters (bo to bk) of the logistic
model are estimated with the use of maximum likelihood method. The probability
of an event to occur can be calculated using the logistic regression model (4)P(Yi=1∖Xi)=ebTXi1+(ebTXi)=11+e−bTXi,   where ebTXi is the linear predictor of the logistic
regression function, and Yi is the event under study (dependent variable).If we use a probability cutoff of.5,
then we can classify an object to group 1 if the estimated P1>.5 and to group 0 if P1<.5. In order to estimate the parameters of the logistic regression
model, the method of maximum likelihood maximizes the coefficients of the
log-likelihood function, a statistic which summarizes the information of the
predictor variables.Both logistic and linear discriminant regression analyses have the same functional frame; a composite of the
independent variables and a rule for classification. But there are many
differences about the assumptions made in order to apply them in a dataset.Regarding discriminant
analysis, the assumptions have great similarity with the assumptions made for ordinary
regression and are (i) independent variables must have a
multivariate normal distribution, thus allowing only continuous or ratio
variables to enter the analysis and excluding all the forms of categorical
variables, (ii) the variance-covariance matrix of all the independent variables
must be homogenic among the population groups divided by the dependent variable
(assumption that is controlled with several statistics, such as Box'sMtest), and (iii) independence of the cases.Accurate estimation of
the discriminant function parameters demands sample size of minimum 20 cases
for each predictor variable and at least 20 cases for each of the dependent
variable groups, otherwise the estimation of the coefficients is unstable and
might lead to misleading results. The dependent variable in a discriminant
analysis should be categorical, dichotomous, or polytomous. The population
groups of the dependent
variable should be mutually exclusive and exhaustive. Discriminant independent
variables are assumed to be continuous. When categorical variables are included
in the analysis, the reliability of discrimination of the analysis decreases [4, 5]. Discriminant analysis is highly sensitive to outliers. Lack of homogeneity
of variances may indicate the presence of outliers in one or more groups. Lack
of homogeneity of variances will mean that significance tests are unreliable,
especially if sample size is small and the split of the dependent variable is
very uneven. Lack of homogeneity of variances and presence of outliers can be
evaluated through scatterplots of variables.Logistic regression also has many
limitations. At first, logistic regression assumes that there is ans-shaped
dependency between the probabilities of group memberships and a linear function
of the predictor variables. It also makes the assumption of independency among
the observations. Analysis of the residuals may reveal patterns that indicate
the presence of multicolinearity or can identify outliers, which can distort
the valid estimation of the logistic coefficients. Also in order for logistic regression
to give trustworthy and reliable estimates, it requires a large number of cases. The more
unequal groups are formed from the dependent variable, the more cases are
needed. On the other hand, logistic
regression does not demand multivariate normality or homoscedasticity for the
predictor variable, but if these conditions are fulfilled, the power of the
prediction is increased [6, 7]. As in OLS regression, outliers can affect results
significantly. The researcher should analyze standardized residuals for outliers
and consider removing them or modeling them separately. Also, unlike OLS
regression, logistic regression uses maximum likelihood estimation (MLE) rather
than ordinary least squares (OLS) to derive parameters. MLE relies on
large-sample asymptotic normality which means that reliability of estimates
declines when there are few cases for each observed combination of independent
variables.For
the evaluation of the two methods, sensitivity, specificity, and accuracy will
be also measured in the same dataset. Sensitivity of a binary classification
test with respect to some class is a
measure of how well this test identifies a condition and expresses the probability
of a case being classified in that class, meaning the proportion of true
positives of all positive cases in the population. Specificity, on the other
hand, expresses the proportion of the true negative classified cases of a
binary classification test of all the negative cases in the population.
Finally, accuracy is a measure of the degree of conformity of a measured or
calculated quantity to the actual value.
It is calculated as the proportion of the true results of a binary
classification test (true positive and true negative) among all possible
results.Thus, linear discriminant analysis
and logistic regression can be used to assess the same research problems. Their
functional form is the same but they differ in the method of the estimation of
their coefficient. Discriminant analysis produces a score, similar to the
production of logit of the logistic regression. Both methods with the
appropriate mathematical calculations produce the predicted probability of the
classification of a case into a group of the dependent variable, and with the
use of the appropriate cutoff value, we can also produce the predicted category
of each observation. When categorical variables are entered in the analysis and
are discrete measured, only the ones with large number of categories, more than
5, approximate the mean and the variance of the variables considered continuous and can be assumed to be normally distributed.
Thus, the assumption of normality is fulfilled, and discriminant analysis makes
robust estimations. On the contrary, logistic regression always produces robust
estimations as it makes no assumption about the distribution of the explanatory
variables or the linear relationship of them with the dependent variable and the equality of the
variance within this group. So, when the
assumptions of the discriminant analysis are violated, we should always avoid
the discriminant analysis and analyze our data with logistic regression, which
gives robust results since it can handle both continuous and categorical
variables [8].


### 2.3. Application

#### 2.3.1. Use of Epidemiologic Data to Evaluate the Prevalence of Asthma

In the following study,
we compared the results of discriminant and logistic regression analyses in
predicting the presence of any asthma symptoms among Greek children aged 10–12 years old
living in urban environment. During 2005, 700 students (323 males and 377 females),
aged 10–12 years (4th–6th grade), were
selected from 18 schools located in several areas of Athens, randomly selected from a list of
schools provided by the regional education offices. The participation rate of
the study was 95%. In order to evaluate
asthma symptoms in the study sample, the parents completed seven questions according
to the ISAAC protocol [9]. Particularly, the evaluation of the presence and the
duration of asthma symptoms was assessed by four questions: (i) if children ever had wheezing,
(ii) if they ever had disturbed sleep
due to wheezing, (iii) if they ever had asthma, and (iv) if they ever had dry
cough at night, except in cases of cold
or chest infection. Further
details about the data used may be found elsewhere [10]. The independent
variables that were associated at a significance level a=0.05 with the
independent variable “presence of any asthma symptoms” were entered in a principal
components analysis (PCA). Eighteen variables were fulfilling the above
criterion, so 18 principal components were extracted from the analysis.
Applying Kaiser's criterion (eigenvalue
>1), we retained 8 factors, all mutually independent.In order to examine the
relationship between childhood asthma and the patterns that are extracted from PCA, the retained 8 components
(patterns) were the predictor variables that entered
in both discriminant and logistic regression models. The assumptions for the two
models were all fulfilled
(the components due to their extraction methods follow the multivariate normal
distribution and are mutually independent), and variance covariance matrices of the groups
were equivalent—Box'sM test of equivalence P-value
>.05—for the
discriminant analysis,
independency of the predictors, absence of multicolinearity after residual
checking, and large number of observations for logistic regression model. We used the standardized canonical discriminant function coefficients
and the unstandardized function coefficients for discriminant analysis and Z
statistic (squared Wald statistic) for logistic regression, to evaluate how much each one of the
variables contributes to the discrimination between two groups. The contribution of the respective
variables to the discrimination depends on how large the coefficients are. We also
compared the sign and magnitude of coefficients. Box's M test was used to check
the equality of the covariance matrices, and it was revealed that they were
equal (P>.05), thus this assumption for discriminant analysis was
met.For each model, we plotted the
corresponding response operating characteristics (ROC) curve. An ROC curve
graphically displays sensitivity and 100% minus specificity (false positive
rate) at several cutoff points. By plotting the ROC curves for two models on
the same axes, one is able to determine which test is better for
classification, namely, that test whose curve encloses the larger area beneath
it. All analyses were performed using the SPSS version 13.0 software (SPSS,
Inc., Chicago, Ill, USA).


## 2.1. Linear Discriminant Analysis

Discriminant analysis focuses on
the association between multiple independent variables and a categorical
dependent variable by forming a composite of the independent variables. This type of multivariate analysis
can determine the extent
of any of the composite variables discriminates between two or more pre-existing groups of
subjects and also can derive a classification model for predicting the group
membership of new observations [1]. The simplest type of discriminant analysis
is when the dependent variable has two groups. In this case, a linear
discriminant function that passes through the means of the two groups (centroids)
can be used to discriminate subjects between the two groups. When there are
more groups, the number of groups minus one function is needed to classify an
observation among them. For each of the groups, linear discriminant analysis assumes the explanatory
variables to be normally distributed with equal covariance matrices. For each case, the estimated
coefficient for an independent variable is multiplied by the case's score on
that variable. These products are summed and added to the constant, and the
result is a composite score, that is, the discriminant score for that case.The
linear discriminant function (LDF) is represented by(1)LDF=b0+b1xi1+b2xi2+⋯+bkxik=bX,   where bj is the
value of the jth coefficient, j=1,…,k, and xij is the value of the ith case of the jth predictor. The
LDF can also be written in standardized form which allows comparing variables
measured on different scales. In the standardized LDF, each variable is
adjusted by subtraction of its mean value and division by its standard deviation.
Coefficients with large absolute values
reflect greater discriminating ability to their corresponding variables. From the LDF, scores can
estimate predicted probabilities and predicted group membership for every case on
the dependent variable. This approach is based on the rationale that it is more
likely that the independent and dependent variables are related as the between-groups sum of square is larger relative to within-group sum of squares. Also the ratio of between-group divided by total sum
of squares (eta-squared statistic or explained variability) or of within-group
divided by total sum of squares (Wilks' lambda statistic or unexplained variability) is used to
assess the relationship. As we can see, the ratio of between-group divided by
within-group sum of squares is an analogue to the ratio of variances, which is
the F statistic, a test that controls the possibility that the observed
relationship is due to chance.The principle by which
the discriminant coefficients (or weights) are selected is that they maximize the
distance between the two group means (centroids)|y¯1−y¯2|.
Fisher [2] was the first who suggested to transform the multivariate
observation x to univariate observations y in such way that the y's
derived from groups 1 and 2 have the maximum distance between them. Thus, the
linear combination y=a′x is the one that maximizes the ratio (squared
distance between sample means)/(sample variance y). The vector of
coefficients is given by the eigenvectors of the matrix B*S−1, where B=(x¯1−x¯2)′ is the between-group matrix and S is an estimate of Σ. A very important characteristic
of these composite sums of squares is that they enclose the variability and the
covariability of each variable. The discriminant coefficients can be calculated
in unstandardized or standardized form but they are irrelatively of the form, less informative than
those in regression. Assuming
that there are 2 groups, x¯1,x¯2 are the means of each group, and S is the pooled covariance
matrix, the allocation rule based on Fisher's discriminant functions is the
following: (2)Xi∈{group1,ify=(x¯1−x¯2)′S−1Xi≥12(x¯1−x¯2)′S−1(x¯1+x¯2),group2,ify=(x¯1−x¯2)′S−1Xi<12(x¯1−x¯2)′S−1(x¯1+x¯2).


## 2.2. Logistic Regression Analysis

Logistic regression is a form of
regression which is used when we want to predict probabilities of the presence
or absence of a particular disease, characteristic, or an outcome in general
based on a set of independent of explanatory variables of any kind (continuous,
discrete, or categorical) [3]. Since the predicted probability must lie between 0 and 1, simple
linear regression techniques are insufficient to achieve that, because they
allow the dependent variable to pass these limits and to produce inconsistent
results.  Defined as P1, the probability of an object is belonging to group 1, and as P0, the probability of an object is belonging to group 0. The logistic
regression model has the form of (3)zi=log(Pi1Pi0)=b0+b1xi1+b2xi2+⋯+bkxik,  where Pi1/Pi0 is called the odds ratio, bj is the value of the jth coefficient, j=1,…,k, and xij is the value of the ith case of the jth predictor.
The parameters (bo to bk) of the logistic
model are estimated with the use of maximum likelihood method. The probability
of an event to occur can be calculated using the logistic regression model (4)P(Yi=1∖Xi)=ebTXi1+(ebTXi)=11+e−bTXi,   where ebTXi is the linear predictor of the logistic
regression function, and Yi is the event under study (dependent variable).If we use a probability cutoff of.5,
then we can classify an object to group 1 if the estimated P1>.5 and to group 0 if P1<.5. In order to estimate the parameters of the logistic regression
model, the method of maximum likelihood maximizes the coefficients of the
log-likelihood function, a statistic which summarizes the information of the
predictor variables.Both logistic and linear discriminant regression analyses have the same functional frame; a composite of the
independent variables and a rule for classification. But there are many
differences about the assumptions made in order to apply them in a dataset.Regarding discriminant
analysis, the assumptions have great similarity with the assumptions made for ordinary
regression and are (i) independent variables must have a
multivariate normal distribution, thus allowing only continuous or ratio
variables to enter the analysis and excluding all the forms of categorical
variables, (ii) the variance-covariance matrix of all the independent variables
must be homogenic among the population groups divided by the dependent variable
(assumption that is controlled with several statistics, such as Box'sMtest), and (iii) independence of the cases.Accurate estimation of
the discriminant function parameters demands sample size of minimum 20 cases
for each predictor variable and at least 20 cases for each of the dependent
variable groups, otherwise the estimation of the coefficients is unstable and
might lead to misleading results. The dependent variable in a discriminant
analysis should be categorical, dichotomous, or polytomous. The population
groups of the dependent
variable should be mutually exclusive and exhaustive. Discriminant independent
variables are assumed to be continuous. When categorical variables are included
in the analysis, the reliability of discrimination of the analysis decreases [4, 5]. Discriminant analysis is highly sensitive to outliers. Lack of homogeneity
of variances may indicate the presence of outliers in one or more groups. Lack
of homogeneity of variances will mean that significance tests are unreliable,
especially if sample size is small and the split of the dependent variable is
very uneven. Lack of homogeneity of variances and presence of outliers can be
evaluated through scatterplots of variables.Logistic regression also has many
limitations. At first, logistic regression assumes that there is ans-shaped
dependency between the probabilities of group memberships and a linear function
of the predictor variables. It also makes the assumption of independency among
the observations. Analysis of the residuals may reveal patterns that indicate
the presence of multicolinearity or can identify outliers, which can distort
the valid estimation of the logistic coefficients. Also in order for logistic regression
to give trustworthy and reliable estimates, it requires a large number of cases. The more
unequal groups are formed from the dependent variable, the more cases are
needed. On the other hand, logistic
regression does not demand multivariate normality or homoscedasticity for the
predictor variable, but if these conditions are fulfilled, the power of the
prediction is increased [6, 7]. As in OLS regression, outliers can affect results
significantly. The researcher should analyze standardized residuals for outliers
and consider removing them or modeling them separately. Also, unlike OLS
regression, logistic regression uses maximum likelihood estimation (MLE) rather
than ordinary least squares (OLS) to derive parameters. MLE relies on
large-sample asymptotic normality which means that reliability of estimates
declines when there are few cases for each observed combination of independent
variables.For
the evaluation of the two methods, sensitivity, specificity, and accuracy will
be also measured in the same dataset. Sensitivity of a binary classification
test with respect to some class is a
measure of how well this test identifies a condition and expresses the probability
of a case being classified in that class, meaning the proportion of true
positives of all positive cases in the population. Specificity, on the other
hand, expresses the proportion of the true negative classified cases of a
binary classification test of all the negative cases in the population.
Finally, accuracy is a measure of the degree of conformity of a measured or
calculated quantity to the actual value.
It is calculated as the proportion of the true results of a binary
classification test (true positive and true negative) among all possible
results.Thus, linear discriminant analysis
and logistic regression can be used to assess the same research problems. Their
functional form is the same but they differ in the method of the estimation of
their coefficient. Discriminant analysis produces a score, similar to the
production of logit of the logistic regression. Both methods with the
appropriate mathematical calculations produce the predicted probability of the
classification of a case into a group of the dependent variable, and with the
use of the appropriate cutoff value, we can also produce the predicted category
of each observation. When categorical variables are entered in the analysis and
are discrete measured, only the ones with large number of categories, more than
5, approximate the mean and the variance of the variables considered continuous and can be assumed to be normally distributed.
Thus, the assumption of normality is fulfilled, and discriminant analysis makes
robust estimations. On the contrary, logistic regression always produces robust
estimations as it makes no assumption about the distribution of the explanatory
variables or the linear relationship of them with the dependent variable and the equality of the
variance within this group. So, when the
assumptions of the discriminant analysis are violated, we should always avoid
the discriminant analysis and analyze our data with logistic regression, which
gives robust results since it can handle both continuous and categorical
variables [8].


## 2.3. Application

### 2.3.1. Use of Epidemiologic Data to Evaluate the Prevalence of Asthma

In the following study,
we compared the results of discriminant and logistic regression analyses in
predicting the presence of any asthma symptoms among Greek children aged 10–12 years old
living in urban environment. During 2005, 700 students (323 males and 377 females),
aged 10–12 years (4th–6th grade), were
selected from 18 schools located in several areas of Athens, randomly selected from a list of
schools provided by the regional education offices. The participation rate of
the study was 95%. In order to evaluate
asthma symptoms in the study sample, the parents completed seven questions according
to the ISAAC protocol [9]. Particularly, the evaluation of the presence and the
duration of asthma symptoms was assessed by four questions: (i) if children ever had wheezing,
(ii) if they ever had disturbed sleep
due to wheezing, (iii) if they ever had asthma, and (iv) if they ever had dry
cough at night, except in cases of cold
or chest infection. Further
details about the data used may be found elsewhere [10]. The independent
variables that were associated at a significance level a=0.05 with the
independent variable “presence of any asthma symptoms” were entered in a principal
components analysis (PCA). Eighteen variables were fulfilling the above
criterion, so 18 principal components were extracted from the analysis.
Applying Kaiser's criterion (eigenvalue
>1), we retained 8 factors, all mutually independent.In order to examine the
relationship between childhood asthma and the patterns that are extracted from PCA, the retained 8 components
(patterns) were the predictor variables that entered
in both discriminant and logistic regression models. The assumptions for the two
models were all fulfilled
(the components due to their extraction methods follow the multivariate normal
distribution and are mutually independent), and variance covariance matrices of the groups
were equivalent—Box'sM test of equivalence P-value
>.05—for the
discriminant analysis,
independency of the predictors, absence of multicolinearity after residual
checking, and large number of observations for logistic regression model. We used the standardized canonical discriminant function coefficients
and the unstandardized function coefficients for discriminant analysis and Z
statistic (squared Wald statistic) for logistic regression, to evaluate how much each one of the
variables contributes to the discrimination between two groups. The contribution of the respective
variables to the discrimination depends on how large the coefficients are. We also
compared the sign and magnitude of coefficients. Box's M test was used to check
the equality of the covariance matrices, and it was revealed that they were
equal (P>.05), thus this assumption for discriminant analysis was
met.For each model, we plotted the
corresponding response operating characteristics (ROC) curve. An ROC curve
graphically displays sensitivity and 100% minus specificity (false positive
rate) at several cutoff points. By plotting the ROC curves for two models on
the same axes, one is able to determine which test is better for
classification, namely, that test whose curve encloses the larger area beneath
it. All analyses were performed using the SPSS version 13.0 software (SPSS,
Inc., Chicago, Ill, USA).


## 2.3.1. Use of Epidemiologic Data to Evaluate the Prevalence of Asthma

In the following study,
we compared the results of discriminant and logistic regression analyses in
predicting the presence of any asthma symptoms among Greek children aged 10–12 years old
living in urban environment. During 2005, 700 students (323 males and 377 females),
aged 10–12 years (4th–6th grade), were
selected from 18 schools located in several areas of Athens, randomly selected from a list of
schools provided by the regional education offices. The participation rate of
the study was 95%. In order to evaluate
asthma symptoms in the study sample, the parents completed seven questions according
to the ISAAC protocol [9]. Particularly, the evaluation of the presence and the
duration of asthma symptoms was assessed by four questions: (i) if children ever had wheezing,
(ii) if they ever had disturbed sleep
due to wheezing, (iii) if they ever had asthma, and (iv) if they ever had dry
cough at night, except in cases of cold
or chest infection. Further
details about the data used may be found elsewhere [10]. The independent
variables that were associated at a significance level a=0.05 with the
independent variable “presence of any asthma symptoms” were entered in a principal
components analysis (PCA). Eighteen variables were fulfilling the above
criterion, so 18 principal components were extracted from the analysis.
Applying Kaiser's criterion (eigenvalue
>1), we retained 8 factors, all mutually independent.In order to examine the
relationship between childhood asthma and the patterns that are extracted from PCA, the retained 8 components
(patterns) were the predictor variables that entered
in both discriminant and logistic regression models. The assumptions for the two
models were all fulfilled
(the components due to their extraction methods follow the multivariate normal
distribution and are mutually independent), and variance covariance matrices of the groups
were equivalent—Box'sM test of equivalence P-value
>.05—for the
discriminant analysis,
independency of the predictors, absence of multicolinearity after residual
checking, and large number of observations for logistic regression model. We used the standardized canonical discriminant function coefficients
and the unstandardized function coefficients for discriminant analysis and Z
statistic (squared Wald statistic) for logistic regression, to evaluate how much each one of the
variables contributes to the discrimination between two groups. The contribution of the respective
variables to the discrimination depends on how large the coefficients are. We also
compared the sign and magnitude of coefficients. Box's M test was used to check
the equality of the covariance matrices, and it was revealed that they were
equal (P>.05), thus this assumption for discriminant analysis was
met.For each model, we plotted the
corresponding response operating characteristics (ROC) curve. An ROC curve
graphically displays sensitivity and 100% minus specificity (false positive
rate) at several cutoff points. By plotting the ROC curves for two models on
the same axes, one is able to determine which test is better for
classification, namely, that test whose curve encloses the larger area beneath
it. All analyses were performed using the SPSS version 13.0 software (SPSS,
Inc., Chicago, Ill, USA).


## 3. Results

Using PCA and applying Kaiser's
criterion, 8 patterns of our original data were extracted, expressing the
anthropometric indexes of the children, breakfast consumption, frequency of
consuming athletic refreshments, parental anthropometric indexes, shortness of
breath during recreational activities, birth weight and breastfeeding, eating
cheese pies, and frequency of listening to music. These variables were used in both discriminant
and logistic regression analyses, and both techniques revealed that
anthropometric characteristics, athletic refreshment consumption frequency, and
eating cheese pies were the most important contributors (Table1). Moreover, we observe
that the direction of the relationships was the same, and there were not
extreme differences in the magnitude of the coefficients. The overall correct
classification rate was 77.4% for discriminant analysis and 79.2% for logistic
regression analysis. Table 2 presents sensitivity, specificity, and accuracy of
both approaches at various cutoffs of the probability of having any asthma
symptoms. Although some differences are observed between the methods, as we can
see in Figure 1, the ROC curves of the aforementioned models clearly indicate
that the logistic model is similar to the discriminant analysis model (i.e., no
difference in the area under the curve (AUC), 74.6% versus 74.4%, P=.9).Table 1
Predictors, standardized,
and unstandardized coefficients for the discriminant analysis model and
logistic regression model.
PredictorsLogistic regressionDiscriminant analysisb coefficientsZ statisticUnstandardized coefficientsStandardized coefficientsAnthropometric characteristics0.5292.6760.3250.319Breakfast eating frequency0.0050.01−0.011−0.011Athletic refreshments frequency consumption−0.6152.784−0.459−0.449Parental BMI0.2681.3970.1030.103Shortness of breath
during activities0.2371.1620.1480.148Birth weight and breastfeeding−0.2891.37−0.182−0.182Cheese pies eating0.3551.6950.2260.225Listening to music
frequency−0.2941.393−0.126−0.126Table 2
Sensitivity and specificity of logistic regression and
discriminant analysis models, at various cutoff points for the probability of
having any asthma symptoms.
Cutoff
value*Logistic regressionDiscriminant analysisSensitivity (%)Specificity (%)Accuracy (%)Sensitivity (%)Specificity (%)Accuracy (%).0594.98.329.61000.825.1.1092.323.340.21001.725.8.256969.269.292.319.237.1.5028.295.879.271.87070.4.755.110076.825.69578.90010075.55.110076.8P* (asthma symptoms): values less than or equal to
the cutoff value indicate that the child is not having any asthma symptoms; those greater than the cutoff value indicate that a child
is having one of asthma
symptoms.Figure 1
Receiver operating characteristics (ROC) curves for
the discriminant analysis and logistic regression models.


## 4. Discussion

In general, both logistic
regression and discriminant analyses converged in similar results. Both methods estimated the same statistical significant coefficients,
with similar effect size and direction, although logistic regression estimated
larger coefficients overall. The overall classification rate for both was good, and either can be
helpful in predicting the possibility of a child having asthma symptoms in the
general population. Logistic regression slightly exceeds discriminant function
in the correct classification rate but the differences in the AUC were
negligibly, thus indicating no discriminating difference between the models.Discriminant analysis can use as a dependent variable a
categorical variable with more than two groups, usually three of four. The number of the predicted
discriminant functions equals with the number of the variable’s categories
minus 1. All of them have different sets of coefficients and produce a
discriminate score for each case, but they have different classification
ability. So, for a four level categorical dependent variable entering
discriminant analysis, three discriminant functions are derived with their correspondent
scores, and only one or two have the necessary power to achieve the optimum
classification rates. The question arises in this case is about the number of
functions which is needed to retain from the available set of functions.In their
paper, Brenn and Arnesen [11] compared the ability of discriminant
analysis, logistic regression, and Cox model when applied in a dataset of 6595 men aged 20–49, who were followed
for 9 years for total and coronary deaths, in order to select possible risk
factors. People in the population
sample were divided into two groups, one with
mortality 5 per 1000 and one with 93 per 1000. Logistic regression and Cox
model derived the same set of variables, and discriminant analysis set of variables
had only minor differences. The researchers also noticed that a time-saving
option, offered for both the logistic and Cox selection, showed no advantage
compared with discriminant analysis, since by analyzing more than 3800
subjects, the logistic and Cox methods consumed, respectively, 80 and 10 times
more computer time than discriminant analysis. Thus, the researchers reached to
the conclusion that discriminant analysis is preferred for preliminary or
stepwise analysis, otherwise Cox
method should be used.In the study of Pohar et al. [8], which used several simulated datasets and
discrimination indexes, the convergence
of the two methods is examined when the linear discriminant assumptions for
normality of the distribution of explanatory variables are met, when they are
violated, and when they are categorized for various parameters of the
explanatory variables such as sample size, covariance matrix, Mahalanobis
distance, and the direction of the distance between the group means. The
authors concluded that linear discriminant analysis is a more appropriate
method when the explanatory variables are normally distributed. For categorized
predictor variables, linear discriminant analysis remains preferable, and logistic
regression overcomes discriminant analysis only when the number of categories
is small (2 or 3). When the assumptions of linear discriminant analysis are not
met, the usage of it is not justified, while logistic regression gives good
results regardless of the distribution of the predictors. In a study by Montgomery et al. [12], who compared the two methods in
veterinary data using stepwise linear discriminant analysis and logistic
regression in a first dataset and comparing the selected variables, the order
of selection and the sign and the magnitude of the estimated coefficients of
the discriminating models in a second dataset, resulted that although both methods converged logistic
regression is preferable to discriminant
analysis
particularly when the assumptions of normality and equal variance are not met.In order to compare the two methods,
we applied them in a real dataset, and we did not use simulation methods, as
the number of the observations in the dataset, although not very large, was
sufficient to provide reliable results. Also, although linear discriminant
function is a better method than logistic regression when the normality
assumptions are met, the differences between them become negligible when the sample size is large enough (50 observations or more).


## 5. Conclusion

In conclusion, logistic regression
and discriminant analyses were similar in the model analysis. In order to
decide which method should be used, we
must consider the assumptions for the application of each one.



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*Source: 952042-2009-03-25.xml*