Based on Notes for Short Course on ‘Random Regression in Animal Breeding’University of Guelph, 19971 INTRODUCTION (J. van der Werf)Multiple trait animal modelAnalysis of individual animal curve parameters3. RANDOM REGRESSION MODELSExample data on stature of four cows (After L.R. Schaeffer, 2001)MODELS FOR GENETIC ANALYSIS OF LONGITUDINAL DATAANS 590AJack DekkersMarch, 2002Based on Notes for Short Course on ‘Random Regression in Animal Breeding’Julius van der WerfLarry SchaefferUniversity of Guelph, 19971 INTRODUCTION (J. van der Werf)In univariate analysis the basic assumption is that a single measurement arisesfrom a single unit (experimental unit). In multivariate analysis, not one measurement buta number of different characteristics are measured from each experimental design, e.g.milk yield, body weight and feed intake of a cow. These measurements are assumed tohave a correlation structure among them. When the same physical quantity is measuredsequentially over time on each experimental unit, we call them repeated measurements,which can be seen as a special form of a multivariate case. Repeated measurements de-serve a special statistical treatment in the sense that their covariance pattern, which has tobe taken into account, is often very structured. Repeated measurements on the same ani-mal are generally more correlated than two measurements on different animals, and thecorrelation between repeated measurements may decrease as the time between them in-creases. Modeling the covariance structure of repeated measurements correctly is of im-portance for drawing correct inference from such data. Measurements that are taken along a trajectory can often be modeled as a functionof the parameters that define that trajectory. The most common example of a trajectory istime, and repeated measurements are taken on a trajectory of time. The term ‘repeatedmeasurement’ can be taken literally in the sense that the measurements can be thought ofas being repeatedly influenced by identical effects, and it is only random noise that causesvariation between them. However, repeatedly measuring a certain characteristic may give1information about the change over time of that characteristic. The function that describessuch a change over time my be of interest since it may help us to understand or explain,or to manipulate how the characteristic changes over time. Common examples in animalproduction are growth curves and lactation curves. Generally, we have therefore two main arguments to take special care when deal-ing with repeated measurements. The first is to achieve statistically correct models thatallow correct inferences from the data. The second argument is to obtain information on atrait that changes gradually over time. Experiments are often set up with repeated measurements to exploit these twofeatures. The prime advantage of longitudinal studies (i.e. with repeated measurementsover time) is its effectiveness for studying change. Notice that the interpretation ofchange may be very different if it is obtained from data across individuals (cross sectionalstudy) or on repeated measures on the same individuals. An example is given by Diggleet al. (1994) where the relationship between reading ability and age is plotted. A firstglance at the data suggests a negative relationship, because older people in the data settended to have had less education. However, repeated observations on individuals showeda clear improvement of reading ability over time. The other advantage of longitudinal studies is that it often increases statisticalpower. The influence of variability across experimental units is canceled if experimentalunits can serve as their own control. Both arguments are very important in animal production as well. A good exampleis the estimation of a growth curve. When weight would be regressed on time on dataacross animals, not only would the resulting growth curve be more inaccurate, but alsothe resulting parameters might be very biased if differences between animals andanimals’ environments were not taken into account. Models that deal with repeated measurements have been often used in animal pro-duction. In dairy cattle, the analysis of multiple lactation records is often considered us-ing a ‘repeatability model’. The typical feature of such a model from the genetic point ofview is that repeated records are thought of expressions of the same trait, that is, the ge-netic correlation between repeated lactation is considered to be equal to unity. Modelsthat include data on individual test days have often used the same assumption. Typically,2genetic evaluation models that use measures of growth do often consider repeated mea-surements as genetically different (but correlated) traits. Weaning weight and yearlingweight in beef cattle are usually analyzed in a multiple trait model. Repeatability models are often used because of simplicity. With severalmeasurements per animal, they require much less computational effort and lessparameters than a multiple trait model. A multiple trait model would often seem morecorrect, since they allow genetic correlations to differ between different measurements.However, covariance matrix for measurements at very many different ages would behighly overparameterised. Also, an unstructured covariance matrix may not be the mostdesirable for repeated measurements that are recorded along a trajectory. As the mean ofmeasurements is a function of time, so also may their covariance structure be. A modelthat allows the covariance between measurements to change gradually over time, andwith the change dependent upon differences between times, can make use of acovariance function. As was stated earlier, repeated measurements can often be used to generateknowledge about the change of a trait over time. Whole families of models have beenespecially designed to describe such changes as regression on time, e.g. lactation curvesand growth curves. The analysis may reveal causes of variation that influence thischange. Parameters that describe typical features of such change, e.g. the slope of agrowth curve, are regressions that may be influenced by feeding levels, environment, orbreeds. There may also be additive genetic variation within breeds for such parameters.One option is then to