Modelling the Effect of Mastitis on Milk Yield in Dairy Cows using Covariance Structures Fitted to Repeated Measures

The data for the present study were compiled from monthly milk yield records of 482 lactating crossbred and frieswal cattle from history-cum-pedigree sheets, daily milk recording registers and disease registers maintained at three Military Dairy Farms at Agra, Bareilly and Lucknow of Uttar Pradesh (India) for the period from 2011 to 2013. The data on milk yield of incomplete lactation were not included in study. Only lactations which complete for eight months were selected.
       
The data were compiled and classified according to various factors such as farm (3 levels), breed (2 levels), calving season (3 levels) and parity (5 levels). Two breeds were considered as crossbred (3/8, 5/8, 3/4, 7/8 and 15/16 of Holstein Friesian and remaining inheritance of other indigenous cows) and Frieswal (62.5% Holstein Friesian and 37.5% Sahiwal inheritance). The calving season was classified as winter (November- February), Summer (March- June) and rainy (July-October). Age of the cows was recorded in years. Monthly milk yield data were available up to 8 months in most of cows, so we have considered only 8 time points in the study. The monthly milk yield data for 9^th and 10^th months were omitted from the analysis to provide balanced data for studying covariance structure of lactation curve. 
       
The mixed model procedure was used to study the effect of mastitis and other factors on monthly milk yield. In the mixed model procedure monthly milk yield up to 8 time points were taken as dependent variable whereas factors such as farm, mastitis, breed, parity and calving season were taken as fixed effects and individual cattle within mastitis group were taken as random effect. Age was taken as covariate in the model.
       
The mixed model methodology for repeated measures data was performed using four covariance structures viz. simple, compound, autoregressive (1) and unstructured (Grohn et al., (1999). Covariance structures were compared using goodness of fit criteria viz., - 2 times the REML log likelihood (-2 R Log L), AIC (Akaike information criterion, Akaike, 1974) and SBC (Schwartz’s Bayesian Criterion, Schwartz, 1978). The smallest value of information criteria indicates a better model fit to the data. Statistical analysis was carried out using proc MIXED procedure in SAS software (Littell et al., 1998).

The prevalence of mastitis and mean monthly milk yield (kg) for lactating cows maintained at three farms are given in Table 1. The overall prevalence of mastitis at three farms was 24.69%. The maximum prevalence was observed at Bareilly farm (27.59%) followed by Lucknow farm (23.36%) and Agra farm (22.81%). The findings were in similar line to those reported by Singh et al., (2014). The prevalence of clinical mastitis in crossbred cows in India ranges between 5% and 37% (Bangar et al., 2016). Contrary to this finding, lower estimates were reported by De and Mukharjee (2009) and Sinha et al., (2014). Chi-square analysis showed that there was no significant (p>0.05) association for prevalence of mastitis at three farms.
 

       
The simple structure showed homogenous variance (13305) among time points without accounting correlations between repeated measures. The estimates of variance and covariance between different time points of lactation under compound symmetry structure were 13489.00 and 9336.64 respectively. The estimate of correlation between different time points of lactation was found to be 0.69. The estimate of variance due to AR (1) structure was observed as 13049. The correlation between any two consecutive time points was 0.84, which decreased as gap between time points increased, with minimum correlation (0.30) between 1^st and 8^th time point. The unstructured covariance matrix has minimum variance (11206) for first time point among all four covariance structure. However, the covariance and correlation pattern was observed decreasing as the length of time interval increases.
       
The parameter estimates of effects of mastitis and other variables on monthly milk yield using four covariance structure is presented in Table 2 and the results indicated that the mastitis causes huge economic loss in dairy cows. While adjusting to other factors such as breed, age, parity, season, farm and month of lactation, it was observed that estimated loss in monthly milk yield due to mastitis were 37.87, 37.99, 35.97 and 36.76 kg under simple, CS, AR (1) and unstructured covariance structure. The simple structure fails to recognize variation between cattle, this result in excessively large F values for breed, age and parity and therefore, this leads to false significance of these factors. But other structures viz., CS, AR (1) and unstructured structure did not show significant effect for these factors. Among other fixed factors, calving season, farms and time points were found significant (p<0.01) for all covariance structures.
 

       
Among all these models, the unstructured covariance had smallest value of -2 R Log L, AIC and SBC. The AR (1) structure, however, had nearly as smaller value of AIC and BIC as unstructured covariance. These finding was in accordance with reports of Grohn et al., (1999). Both of these structures fit better than simple and CS structure. Based on goodness of fit measures, it was concluded that unstructured covariance structure was best for modeling of the effect of mastitis on monthly milk yield of dairy cows. These findings are in agreement with reports of Akbas (2002) and Shukla and Kumar (2012). Contrary to this finding, Littell et al., (2000) found that autoregressive with random effect is the best choice of covariance structure. Also, Grohn et al., (1999) and Wilson et al., (2004) preferred first order autoregressive structure over unstructured covariance structure to study the effect of disease on milk yield in dairy cows.

Mixed modelling using four covariance structures was done to estimate the effect of mastitis on milk yield by accounting the correlations between repeated measure data of monthly milk yield in dairy cows. The simple covariance showed false significance of some variables, however, by accounting the correlation in varied degree, CS, AR(1) and Unstructured covariance structure showed reliable results. On the basis of goodness fit measures, it was concluded that unstructured covariance structure was found to more appropriate method for better fitting of the repeated measures data and it was estimated that the mastitis caused 36.76 kg losses in monthly milk yield in dairy cows.

The authors are highly thankful to the Deputy Director, General of Military Farms for their consent to use the data for the present study. The authors are also thankful to learned referees and the Editor for their valuable comments on the original version of the paper.