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ANIMAL GENETICS |
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* Department of Animal Sciences, University of Wisconsin, Madison 53706;
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Department of Dairy Science, University of Wisconsin, Madison 53706;
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Department of Biostatistics and Medical Informatics, University of Wisconsin, Madison 53706;
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Department of Animal and Aquacultural Sciences, Norwegian University of Life Sciences, N-1432 Ås, Norway;
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# Institute of Agricultural Biology and Biotechnology (IBBA), National Research Council, Milan 20133, Italy;
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|| Department of Animal Pathology, Hygiene and Veterinary Public Health, University of Milan, via Celoria 10, 20133 Milan, Italy
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Abstract |
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 Top Abstract INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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Key Words: goat • intramammary infection • structural equation model • simultaneity of effects
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INTRODUCTION |
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TopAbstractINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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; Zeng and Escobar, 1995; Rodriguez-Zas et al., 2000). It is not clear if this association is due to a dilution effect of MY on the concentration of somatic cells in milk, denoted as MY
SCC, or if it reflects an effect of disease on production; i.e., an effect from SCC on MY, denoted as SCCMY, or if it is due to common (genetic or environmental) factors, or if it is due to a combination of all these causes.
A better understanding of the dynamics of intramammary infections in dairy goats might be obtained by analyzing records collected from each udder half. Moroni et al. (2005b) reported an increase in SCS of uninfected udder halves of infected goats. If the immune response to infection is not local, a simultaneous relationship between SCC of the right and SCC of the left halves of the udder may be postulated. Also, increased levels of SCC can be observed even after infection disappears. If there is a time lag before SCC returns to a normal level after infection, an autoregressive process within halves over test days may be needed for describing longitudinal records of SCC.
Standard linear model theory does not accommodate the recursive (i.e., an effect from one response variable onto another response) and bidirectional effects (the case where response Y1 appears in the equation of response Y2, and vice versa) discussed above (Gianola and Sorensen, 2004). However, Structural Equation Models (StEM) allows modeling of such features. Here, the LISREL (Linear Structural Relationships, Jöreskog and Sörbom, 2003) software was used for fitting several StEM to somatic cell score (SCS = log2[SCC + 1]) and MY records of dairy goats.
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MATERIALS AND METHODS |
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TopAbstractINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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described the sampling and laboratory procedures followed in the program. Within the 47 goats, 20, 14, and 13 were from first, second, and third (or greater) parity, respectively. Average days in milk at tests 1, 2, 3, and 4 were 46, 65, 93, and 120, respectively, with a SD of 13 d. All variables were expressed as deviates from the appropriate test-day mean. Following Ali and Shook (1982), SCC was transformed into a SCS.
Models
In the StEM fitted here, MY and SCS were viewed as response variables, parity and infection status were observable predictors, and goat was treated as an unobservable random effect. Figure 1 shows the path diagram of a model allowing for simultaneity of effects between SCS and MY at each test-day, simultaneous effects of SCS between halves within test-day, and an autoregressive process for SCS within half. In the diagram, a single-headed arrow indicates a linear effect of the variable at the tail of the arrow on the variable at the head of the arrow; double-headed arrows indicate covariances. Two arrows pointing in opposite directions indicate simultaneous effects between variables involved in the bidirectional flow. For clarity in the figure, subject subscripts, residuals, and intercepts (peculiar to each test day) were omitted.
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, test-day MY records were affected by test-day-specific intercepts, the random animal effect (AMY, assumed to be the same for all test-days), by parity of the goat (P2 and P > 2 are 2 dummy variable indicators of effects of second and later parities, respectively), and by SCSR and SCSL at the corresponding test-day. The SCSR and SCSL were also supposed to be affected by test-day, animal (ASCS, assumed to be the same for all tests and halves), by infection status of the corresponding half (hereinafter BR or BL stand for bacteriological status of the right and left half, respectively) at the same test-day, by an effect of SCS of the other half at the same test day (SCSR
SCSL, where denotes simultaneity of effects of variables at the end of the arrows), and by a within-half autoregressive effect of the SCS in the preceding test-day. A covariance between goat effects on SCS and on MY was allowed.
Equation 1 gives the matrix representation of the model in Figure 1 for records of goat i (i = 1,2,...,47):
 | [1] |
where myi = (my1i, my2i, my3i, my4i)' and scsi = (scsr1i, scsl1i, scsr2i, scsl2i, scsr3i, scsl3i, scsr4i, scsl4i)' are vectors of MY and SCS records on subject i, respectively; 
= (MY1,..., MY4, SCS1, SCS1,..., SCS4, SCS4)' is a vector of test-day intercepts of MY and SCS;
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is a matrix defining simultaneous or recursive effects, where ß1 denotes the effect of the SCS from a right or left half on MY recorded at the same test-day; ß2 is the effect of MY on SCS observed in the right or left half at the same test-day; ß3 is the effect of SCS for one-half on the SCS of the other half; and ß4 is the effect of SCS in a given half at test-day t on SCS of the same half at test-day t+1. Further, in Eq. [1], 
is a matrix defining direct effects of predictors on response; ß5 and ß6 are the effects of second and third or later parity, respectively, on MY; and ß7 is the effect of infection status of a given test-day and half on the SCS measured on the same half at the same test day. The vector xi = (p2i, p > 2i, b1ri, b1li,..., b4ri, b4li)' in Eq. [1] contains parity class and infection status variables. Note that, under this parameterization, intercepts are means of MY or SCS of first parity dairy goats, free of infection. Further, in Eq. [1] ai = (amyi, ascsi)' is the vector of goat effects on MY and on SCS, respectively, and Z is the appropriate incidence matrix for the goat effects. Finally, 
i = (MY1i,..., MY4i, SCSR1i, SCSL1i,..., SCSR4i, SCSL4i)' is a 12 x 1 vector of model residuals.
Multivariate normality was assumed for the vector of all predictor variables (observable and unobservable). This assumption is violated because some of the predictors (parity, infection status) are discrete; however, the assumption is required in LISREL. Thus,
 | [2] |
In Eq. [2], µx is the mean of x; 
is an unrestricted positive-definite (co)variance matrix; G is the 2 x 2 variance covariance matrix of goat effects; and 
is the residual (co)variance matrix, where 
MYk2, SCSk2 are residual variances of MY and SCS at test-day k, and 
denotes the Kronecker product. Note that allowance was made for heterogeneous residual variances of MY and SCS across test periods, with homogeneous variance assumed for SCS of right and left halves within a test-day. With the assumptions in Eq. [1] and [2], the likelihood is multivariate normal with a structured variance-covariance matrix and mean vector. The set of parameters entering in the structured mean vector and (co)variance matrix are the distinct elements of , G, R, , and ß = ß1, ß2,..., ß7)'. The LISREL produces maximum likelihood estimates of these parameters (Jöreskog and Sörbom, 2003).
Table 1 gives the effects included in each of 8 models fitted. In the null model (Mnull), MY and SCS were affected only by goat and residual effects, so this is a random effects model. In M0, exogenous effects of parity and infection status were added. Models with a single recursive effect were: M1a, with a recursive effect of SCS on MY (SCSMY); M1b, with the opposite effect (MYSCS); M1c, with simultaneity in SCS between halves within test-day (SCSRSCSL); and M1d, which included a first-order autoregressive process in SCS within half (SCStSCSt+1), where t denotes a test-day period. Based on evidence provided by the Bayesian information criterion (BIC, Schwartz, 1978), as discussed later, 2 additional models were fitted: M2 including both SCSMY and SCStSCSt+i, and M3 which was M2 plus within-test day simultaneity of effects of SCS of the 2 halves (SCSRiSCSLi). Finally, a full model (Mfull) including all effects in Eq. [1] was fitted.
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), which provides an approximation (on a logarithmic scale) to the Bayes Factor for 2 models (ratio of model posterior probabilities) when the 2 models have equal prior probabilities. Each model was compared with the saturated model, this being a model in which the (co)variance matrix and mean vector of observable variables is completely unstructured. With p observable variables, the saturated model has p(p+1)/2 + p parameters. Here, p = 22, consisting of 4 MY records, 8 SCS, 2 dummy variables coding parity, and 8 binary indicators denoting infection status for the appropriate udder halves. The model being fitted was the restricted model. The degrees of freedom of the 
2 statistic is the difference in number of parameters between the saturated and fitted models. Any pairs of models fitted can be compared using BIC differences. Raftery (1995) indicates that BIC differences of 2, 6, and 10 can be considered as positive, strong, or definite evidence in favor of the model with smaller BIC, respectively. For nested models, and using the restricted model as null hypothesis, these BIC values can be associated with P values of 0.0198, 0.0030, and 0.0005, respectively. These values were used as references to evaluate observed BIC differences.
Direct and Total Effects
In a StEM, the total effect of a variable on another variable can be decomposed into direct and indirect effects. A detailed discussion about this decomposition of effects and of the related problem of spurious correlation is given by Alwing and Hauser (1975). To illustrate, consider a bivariate model in which Y1 = ß1X1 + 1 and Y2 = ß2X1 + ß3Y1 + 2; this system implies that Y2 = (ß2 + ß3ß1)X1 + ß31 + 2. The direct effect of X1 on Y2 is ß2, and the total effect is (ß2 + ß3ß1). More generally, the total effect of a set of variables can be evaluated based on the reduced form of the model. For instance, for Eq. [1], the reduced form model is
 | [3] |
Therefore, the matrix of total effects is
 | [4] |
Variance Components
Estimates of (co)variance components were obtained for each of the models fitted. To compute these components, one needs to consider the reduced model (e.g., Gianola and Sorensen, 2004). From Eq. [2] and [3], the phenotypic (co)variance matrix is
 | [5] |
Replacing parameters in the right hand side of Eq. [5] by its estimates leads to an estimate of the phenotypic (co)variance matrix.
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RESULTS |
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TopAbstractINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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Table 1 shows the 2, –2(log-likelihood), and BIC statistics for each model. There was strong evidence supporting the existence of a recursive effect of SCS on MY and of a within-half first-order autoregressive process for SCS. When these effects were fitted jointly (M2), adding an effect from MY on SCS and reciprocity between SCS of different halves (Mfull) improved the fit of the model, but BIC did not change appreciably. The improvement in fit was not enough to overcome the penalty against additional parameters imposed by BIC.
Table 2 displays estimates of effects, by model. Estimates of effects of explanatory X-variables on responses were fairly similar across models. A positive bacteriological test was associated with an increase in SCS. The MY was about 0.4 kg/d larger in second than in first lactation goats and about 0.57 kg/d larger in third or later parity goats than in first lactation animals. The SCS had a negative effect on MY of about 16 to 20 g/d per unit of SCS. The first-order lagged effect on SCS, ß4, was positive (implying carryover of infection), and the reciprocal effect on SCS of different halves, ß3, was positive as well, implying that immune response is not strictly local. The estimate of the effect from MY on SCS (dilution) was negative (–1.236 and –0.598 SCS/ kg in M1b and Mfull, respectively), but not significantly different from 0 at the 1% level. Given the small number of goats in this study, this estimate only suggests a possible existence of a dilution effect. However, comparison between M1b and Mfull indicates that the dilution effect becomes smaller after carry over effects and simultaneity between halves enter into the model.
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gives the estimate of the matrix of total effects, in the sense of Eq. 4, for model Mfull. On average, second parity goats are expected to produce 0.43 kg of milk per day more than first lactation goats, whereas those from third or later lactations would produce 0.59 extra kg per day than first lactation goats. In all models the direct effect of parity on MY was restricted to be equal across test-days; however, because of indirect effects of parity on MY through SCS and through the lagged effect of SCS of previous tests-periods, the total effect of parity on MY is actually test-day specific, as given in Table 3. Similarly, although the model did not postulate a direct effect from parity on SCS, there was an indirect effect mediated by MY, as shown in Table 3. Conditionally on infection status, SCS is expected to decrease as parity increases, and this would be a consequence of the dilution effect. Because of the autoregressive process assumed in Mfull, the effect of parity on SCS was larger in absolute value at later stages of lactation. Infection in both halves (BR = 1 or BL = 1) tended to reduce MY by as much as 30 g at the test-period at which infection was present, and the effect tended to dissipate in subsequent periods. Infection in one half of the udder would be expected to increase SCS in the corresponding test day by 1.574 units in the half where infection occurred and by 0.194 units in the other half. According to these estimates, infection is expected to affect SCS in the following test periods due to the lagged effect.
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DISCUSSION |
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TopAbstractINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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SCS) was not clear. In order to avoid possible biases induced by omitted effects, discussion is based on estimates from Mfull.
Effects of Predictor Variables on Responses
Infection incidence was moderately high (20%), but this is not unusual in goats. Similar infection rates were reported by Sánchez et al. (1999), who found 22 and 37% half and animal-level prevalence, respectively (324 goats), and by Corrales et al. (1996), who found a 17% half-level prevalence (1,206 halves). Contreras et al. (1996), analyzing 1,892 milk samples from 160 Murciano-Granadino goats, reported a 10% infection rate.
Estimates of the effect of infection on SCS varied across models. In Mfull, it was 1.53 units, whereas in other models (e.g., M1a) the estimates were close to 1.8 units. Estimates reported here appear to be larger than reported elsewhere. Kosev et al. (1996), analyzing 3,040 milk samples, found an effect of an infection of 2 SCS units. Contreras et al. (1996) and Corrales et al. (1996) reported estimates of 0.86 and 0.82 units, respectively. Variation in estimates may be partially due to differences in type of pathogens between studies. Eitam and Eitam (1996) and Corrales et al. (1996) provided estimates of effects of different pathogens on SCC in goats. Gonzalo et al. (2002) estimated effects of different pathogens in sheep in a study involving 1,332 ewes; in a log10 scale, estimates varied from 1.56 (Streptococcus agalactiae) to 0.08 units (Micrococcus spp.). In the SCC scale, the estimate obtained here of about 1.53 units of SCS suggests that, from a preinfection SCC level of 50(300) thousand cells/mL, an infection would increase SCC to 144(866) thousand cells/mL, computed as 50 x 21.53 or 300 x 21.53, respectively.
In models in which a recursive effect of SCS on MY was fitted, infection had an indirect effect (mediated by SCS) on MY. Results from Mfull predict that, if one-half is infected, MY is expected to decrease by about 30 g per day. Infection in both halves would reduce expected MY by 60 g. Also, infection in consecutive test periods would reduce expected MY even further, through the lagged effect on SCS. There are few references on effects of subclinical mastitis on MY. Peris et al. (1996) reported an effect of udder infection on potential milk yield (estimated after double injection of oxytocin) in Manchega ewes of –243 g/d. In a larger scale study, Gonzalo et al. (2002) found effects of infection on MY of –89 and –77 g/d for bilateral and unilateral infections, respectively. However, Moroni et al. (2005b) found no effect of infection on MY. In this study, the estimate may be biased downwards because MY and SSC were not collected on the same day.
The estimate of the effect of parity on MY was stable across models. Using estimates from Mfull, goats of second and third parities or greater are expected to produce 0.43 and 0.59 more kg of milk per d, respectively, than first parity goats. Rota et al. (1993) reported similar estimates (0.51 and 0.55) in a study with 50 goats. Mavrogenis and Papachristoforou (2000) analyzed 1,611 lactations of 486 Damascus goats and reported significant effects of parity on MY but did not present estimates. In models including an effect from MY on SCS, the effect of parity on SCS is mediated by the recursive effect MYSCS. An increase in parity number would increase expected MY, and because the estimate of MY on SCS was negative, it would reduce SCS. On the other hand, most reports indicate that SCS tends to increase with parity number (Rota et al., 1993; Aleandri et al., 1996; Contreras et al., 1996). A distinct aspect of the StEM presented here is that infection was included as a predictor. In other words, the estimate of the total effect of parity on SCS is at a fixed infection status, and it reflects the dilution effect caused by increased production. Moroni et al. (2005a) reported that goats from third and fourth parity were more likely to get an infection than goats from either first or second parities. Thus, the observation that SCS tends to increase with parity may be due to a more complex process mediated by recursive effects of infection and of MY on SCS.
First Order Autoregressive Process for SCS
An increase of 1 SCS unit at test i in the right (left) half would increase expected SCS in test i+1 in the right (left) half by 0.243 units (tests were approximately 1 mo apart). The lagged effect may be capturing a false negative infection in test i+1 (assuming that false negatives are more likely if SCS was high in the previous test), or perhaps immune response does not dissipate in a 30-d period. de Haas (2003), analyzing longitudinal records in dairy cows, described somatic cell patterns observed under infection by different pathogens. In that study some pathogens, especially Staphylococcus aureus, induced increased SCC for a long period after infection.
Simultaneity Between SCC and MY
A negative covariance between raw records of SCS and MY has been reported in goats. Zeng and Escobar (1995) found a correlation of –0.46 (116 samples taken on 15 goats). Wilson et al. (1995), analyzing records from 117 goats (8 repeated records per animal), found a negative effect of 305-d MY (treated as fixed) on SCS. Rota et al. (1993) noted that lactation curves of the 2 traits followed opposite trajectories (high SCC coincides with low MY, and vice versa). The covariance between SCS and MY can be explained partially by simultaneity of effects between traits. The evidence in favor of a negative effect from SCS to MY seems stronger than the evidence in favor of a dilution effect (MYSCS). However, evidence in favor of a dilution effect was suggestive. Simultaneous effects have an impact on the phenotypic covariance between records. Here, the 2 estimated simultaneous effects were negative, which induces a negative phenotypic correlation between SCS and MY.
Simultaneity in SCS Between Halves
Results suggested existence of reciprocal effects between halves. This could reflect that immune response is not strictly local. However, perhaps part of the effect reflects bias due to false negative cases of infection, assuming that a false negative in a given half is more likely to occur if the other half has a high SCS. The estimate of the reciprocal effect in Mfull (0.119) was moderate, and it indicates that an increase in SCS is expected to elevate SCS moderately in the other half. Peris et al. (1996), analyzing records from 72 Manchega ewes, presented means of SCS, by half, in ewes with mastitis in one half and in healthy ewes. Average SCS of infected and healthy halves of mastitic ewes, and of halves of healthy ewes was 6.55, 4.78, and 4.68, respectively. Mean SCS of healthy halves of mastitic ewes was somewhat larger than in healthy ewes, but the difference was not significant. Results presented here are also consistent with those of Moroni et al. (2005b) who reported that uninfected udder halves of infected goats had lower SCS than infected halves of the same goat but higher SCS than that from goats that were free from infection.
Conclusions
The SCS affects MY negatively, and this may reflect the negative effect of disease, measured indirectly by SCS, on yield. Evidence in favor of a dilution effect of MY on SCS was not clear, and further studies with larger sample size are needed to arrive at more solid conclusions in that respect. The highly significant first-order autoregressive process on SCS indicates that immune response to infection does not dissipate in a short period. Immune reaction to infection has a strong local component, expressed as an increase of SCS in infected halves. However, there was evidence of an animal-level immune reaction because there was an increase of SCC in milk from a healthy half when the other half was infected.
Future studies, involving larger data sets, and relaxation of some of the assumptions (e.g., treating infection as endogenous, and parity as fixed, or using other instruments different than CM and parity) are needed to explore some of the suggestive findings of this study in more depth.
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Footnotes |
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2 Corresponding author: gdeloscampos{at}wisc.edu
Received for publication January 7, 2006. Accepted for publication June 26, 2006.
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LITERATURE CITED |
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TopAbstractINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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