Logistic Regression (LR) has been studied recently in different scientific fields such as medicine, economy and agriculture. LR is a multivariate technique where dependent variable is only categorical (generally binary) and independent variables, mixtures of categorical and continuous variables. LR can be prepared due to some reasons: there are no assumptions on distributions of independent variables, residual terms and on heterogeneity of variances.
Considered on animal science, effects of environmental factors such as sex, birth type, birth weight, genotypes and dam age on different growth periods are important.
This study aimed to examine the effects of sex, birth type and birth weight on body weights at different periods by using LR instead of variance analysis and to assess model fit criteria such as Chi-square test G test and concordant test using for LR.
MATERIALS AND METHODS
Animal materials of this study were composed of 45 Hamdani lambs raised in Van province of Turkey. Data of sex (26 male and 19 female), birth type (32 single and 13 twin) a nd body weights at different periods (birth weight, body weights at 45th, 60th and 75th days) of the lambs born in early March of 2001 year were recorded.
Body weights at these periods categorized as binary (low and high) are presented in Table 1. Male sex and single birth type were determined as reference categorizes of independent variables in this study. Data obtained from this study were analyzed with procedure freq and logistic of SAS program.
Logistic regression is a part of generalized linear model. Dependent variable consists of data coded as binary (0 and 1) (Table 2) while independent variables compose of grouped or categorized data (0, 1, 2) and continuous data. Category coded as 1 in dependent variable, called as reference category, is risk factor of interest.
The LR model can explain the effects of independents variables on binary dependent variable.
Where, β0 is the intercept parameter and β is the vector of slope parameters.
|| Categorized ranges of body weights at different periods
|| Cross tabulation of variables
Occurrence probability of reference category in dependent variable, denoted by P (Y=1), can be written as follows:
P(Y=0), other category of dependent variable, can be also written as:
Odds ratio (O), the ratio of Eq. 2 and 3, are given in Eq. 4 :
After transformed to log-scale, Eq. 4 can be rewritten as follow:
It can be said, therefore, the Multiple Regression Analysis (MRA) is analogous to logistic regression as seen in Eq. 5[1,5].
The association between odds ratio (O) and the probability of being P (Y=1) can be expressed as:
Where, P is the probability of risk occurrence, denoted normally by P (Y=1) and O is odds ratio.
The odds of being high weight in terms of body weight at 75th day for male were 3.818 times higher than that for female.
Model fit criteria: Likelihood ratio chi-square (called also as G-test)
(Eq. 7) and chi-square statistics (Eq.8),
generally known as goodness of fit criteria, are used for assessing whether
model fits or not in logistic regression[4-6]. Expressions of both
statistics, also used for many multivariate statistics techniques, can be briefly
written as follows:
Where, f observed frequency and f1 ; expected frequency, respectively.
Concordant: The criterion is a measure of association between observed responses and the predicted probabilities.
RESULTS AND DISCUSSION
It was observed in Table 3 that associations of birth type only with body weight at 30th day and 60th day (p<0.01) as well as association of sex only with body weight at 30th day were significant (p<0.05). As examining Table 4, associations of birth weight with body weight at 30th day, 60th day and 75th day, respectively, were much significant (p<0.01).
As examining Table 5, odds of being high weight at 75th day for male was 3.818 times higher than that for female (p<0.05).
Odds of being high weight at 75th day for being single birth type was 8.038 times higher than that for being twin birth type (p<0.05).
|| Associations of sex and birth type with body weights at different
|| Associations of birth weight with body weights at different
|*:p<0.05, **: p<0.01
|| Individual effects of sex birth type and birth weight on
body weights at different periods.
|*: p<0.05, **: p<0.01
Odds of being high weight at 60th day for being single birth type was 15.429 times higher than that for being twin birth type (p<0.05). It could be said that lambs of being single birth type had higher weight with respect to others. Examining in concordant test, known also as model fit criteria except G and Chi-Square tests in LR, additions of factors such as sex a nd birth type as independent variables on model fit had low-moderate level (28.4-43.7). On others hand, addition of birth weight in model on model fit had also high level with respect to sex birth type (about 77).
It is known that effects of sex, birth type and birth weight on body weights at different periods are important. This study was to interpret odds ratio in cross tabulation and to examine in model fit criteria such as G, Chi-Square and Concordant in LR.
Birth type on body weights at 45th day and 60th day in terms of G test and Chi-square test were effective. Odds of being high weight at 45th and 60th days for single birth type were 8.038 and 15.429 times than those of twin birth type, respectively (p<0.05) (Table 4). Addition of the factor on model fit was moderate level.
Effect of sex on only body weights at 75th day was significant (p<0.05)
(Table 3 and 5). Odds of being high weight
at the periods for male lambs were 3.818 times than that for female lambs.
With respect to results of G test and Chi-square test, relationships between birth weight with respectively body weight at 30th day, 60th day and 75th day were much significant (p<0.01).
Contrary to other factors, birth weight factor improved model fit at all periods (Table 3 and 4). When a unit increasing in Birth weight realized, odds of being high weight at those periods were much high and significant (Table 5).
As a result, it can be suggested that calculation of Chi-Square test with G test in cross tabulations should be needed. With those tests, both using as model fit criteria in LR and proving relationship between two traits are possible. Besides, researchers may obtain much information by LR instead of variance analysis.