1. Introduction

There are many factors influencing the fertility in dairy cattle. Days open is one of these factors which play an important role in reproductive and productive efficiency of individual cows (Arthur et al., 2001). Days open is the number of days between the most recent calving and conception (calving conception interval). Standard statistical techniques such as multivariable regression analyses used to describe and quantify the effect of factors influencing days open in dairy cattle. Survival analysis is an important area in statistics as it gives a number of advantages over other regression techniques that it can solve the problem of censored observations (Kleinbaum and Klein, 2005). Censoring means that the animal does not achieve the expected event during the study period. Datasets in survival analysis are usually non-negative, continuous and its specific survival function. A survival function can be detected by adjusting parametric models, which is usually nonlinear models, instead of the usual non-parametric (Kaplan-Meier) and Cox proportional hazard model which is a semi-parametric (Colosimo and Giolo, 2006).

The use of censored data provides less biased estimates for factors affecting reproductive performance (Del et al., 2005; Del et al., 2006) and for this reason, survival analysis is the recommended method for analysis of dairy cow reproductive data.

Since survival data are usually asymmetric, the assumption of normality of the data is not essential. Therefore, exponential, normal, log-normal, Weibull, logistic and log logistic distributions are generally used for these data (Collet and Kimber, 2011).

2. Material and Methods

Data for present study were obtained from animal records belongs to different Dakhalia farms (n = 1842) Covering the period between 2009 and 2011. The data set were fitted to different survival parametric models. The animals which became pregnant at the study period are considered complete data.

The following categories of records were considered as censored data:

Ÿ Cows still non pregnant at the end of the study period;

Ÿ Cows sold to anther herd.

Ÿ Cows culled from the herd throughout the breeding period for reasons apart from reproductive failure.

Ÿ Cows with days open, or interval between last calving and the end of data, longer the study period were considered censored (Roxstrom et al., 2003).

Variables under study were days open is a dependent variable. Season of calving (categorical independent variable) is coded as (1) in case of summer season and (summer season is omitted) and (2) in case of winter season. Calving interval, previous dry period and age at calving are continuous independent variables. Lactation order (categorical independent variable) is coded as: lactation order 1= (1), lactation orders 2 = (2), lactation order 3 = (3), lactation order 4 = (4) and lactation order 5 and over = (5). (Lactation order 5 and over is omitted as a reference group).

2.1. Statistical Data Analysis

Multivariate regression or parametric survival analysis was performed using several survival modeling methods. Several accelerated failure time parametric models are exponential, normal, log-normal, Weibull, logistic, log logistic and smallest extreme value distributions. All survival models were constructed using life data regression procedure of Statgraphics statistical package program (Statgraphics, 2015). Estimates for the parameters from different models fits estimated by using maximum likelihood and comparisons among different models were done by using Akaike information criterion (AIC) (Akaike, 1974).

Lower AIC values indicate a better model fit. The formula for AIC is:

AIC = -2 log L + 2k

where log L is the log likelihood of the proposed model, and k is the number of model parameters.

As it is known that the core of survival analysis is survival and hazard functions. These functions are used in the process of prediction of the event under study.

Survival function, density function and hazard function for different parametric models were shown in Table (1) as they are the core of survival analysis.

Table (1). Survival function, density function and hazard function for different parametric models

In the above models, s(t) is the survival over time (per days); λ is a scale parameter associated with the average lifetime in the exponential model. γ is a shape parameter associated with the hazard ratio; and µ and σ are scale and location parameters in case of hazard and density function. β also is the 3% scale parameter in case of smallest extreme value distribution (Colosimo and Giolo, 2006; Lee and Wang, 2003).

2.1.1. Parametric Survival Model

The parametric regression model can also be used to test the relationship between the time variable (days open) and the explanatory variables. Generally any distribution defined for t ∈[0,∞] can be treated as a survival distribution. Through a logarithm transform we get log (t) ∈(−∞,+∞) which is more suitable for modeling. For example, a Weibull distribution which is most commonly used in survival analysis. The general regression model is given below (Rodriguez, 2005).

Where 𝛼 = −log (h(𝑡)) has a specific distribution.

For Weibull distribution,

Where,

X₁,…, are explanatory variables

B₁,…, are regression coefficients

σ is a scale parameter

W has the extreme value distribution

U is an intercept

The regression parameters will be used to interpret the direction and strength of the relationship of each explanatory on the effect of the survival time. Positive value of B_p is indicative of increased survival.

3. Results

A total of 1842 animals divided into 854 uncensored values with percent 46.36% (pregnant animals) and 988 censored values with percent 53.64% (non-pregnant).

After constructing different types of parametric survival models Weibull, normal, exponential, logistic, log-logistic, lognormal and smallest extreme value and compared their AIC (Table 2) to determine the best fit model. The Weibull model had the lowest AIC which is (10956.42) in days open dataset, indicating a better overall fit than the other models with parameters 5.2907(0.0901337). The Logistic was not appropriate to describe the dataset.

The parameters from different model fits are displayed in Table (2). Positive value of is indicative of increased survival (increasing the event occurrence). The Akaike Informational Criterion (AIC) is also provided at this table.

Table (2). The parameter estimates and Akaike Informational Criterion (AIC) of different models

Generally, positive values for a beta coefficient of different variables indicate a better survival and this means that the cow has a good chance to be pregnant, and negative values give worse survival (pregnancy chance is lower) according to Gamel and McLean (1994).

Because the Weibull model had the lowest AIC score and demonstrated a good fit of the data, so this parametric model was selected as the best model.

The equation of the Weibull fitted model is

Days open = exp (5.2907 + 0.000297543 * Age at calving - 0.000875385 * Dry period + 0.000517566 * Calving interval + 0.0348342 * Season (1) + 4.30334 * Lactation order (1) + 0.0305605 * Lactation order (2) + 0.0764563 * Lactation order (3) + 0.0853323 * Lactation order (4).

As it is known that positive values for a beta coefficient depict better survival, and negative values depict worse survival. For example, the negative beta coefficient (- 0.000875385) for a cow indicates that these cows are observed to have a worse chance of pregnancy compared to others with positive beta coefficients.

Variables such as age at calving, calving interval and season are associated with markedly improved survival as their beta^’s values are (+ 0.000297543, + 0.000517566 and + 0.0348342 respectively). Increasing in lactation orders variable may seem to imply a better chance of pregnancy for these cows due to positive values for beta coefficients as in Table (3).

Table (3). The beta coefficients for the Weibull survival model

This Weibull equation can be used to predict pregnancy in cows depending model parameters and beta coefficients of the explanatory variables. The time of pregnancy (years or months) for each cow can be detected in each lactation order and season.

4. Conclusions

The parametric survival models can be used to model dairy data as it deals with censored values. The AIC is used to evaluate model performance, as this is one of the most widely used metrics. The Weibull model demonstrated the best performance compared to other parametric models so its equation is used in predicting process of the event under study. The Logistic was not appropriate to describe the dataset. Finally, this model is good for prediction process of all events belonged to dairy farms depending on the variables affecting these events.