This study was designed to explore the efficiency of mixed model over fixed effect model.rnFurthermore, it was designed to determine how the REML procedure was used to find thernestimate of variance-covariance matrix of the model. The methods of restricted maximumrnlikelihood are applied to the data from uniformity and multi-location trials. In this study,rntwo data sets are used; uniformity trial which was used to determine optimum plot sizernand shape, and multi-location trial which is used to identify the effect of location onrndifferent varieties of wheat.rnIn uniformity trial, two sowing methods: row and broadcast sowing, are considered bothrnin 1996 and 1997 in which the trial were conducted. These factors are considered as fixedrneffect of the model. But, plot sizes ranging between 1 and 24 m2 with different shape arerntaken as random factors of the model.rnRegarding Multi-location trial, it was conducted in 2001, 2002 and 2003 using 20 wheatrnvarieties in 14 locations that were expected to have the same agro-ecological effect forrnwheat adaptations. All the three years in which the trail was conducted are taken as fixedrneffect whereas varieties and locations are considered as random factors of the model.rnThe result of the study based on restricted maximum likelihood (REML) revealed thatrnsmall plot sizes have larger coefficient of variation in both 1996 and 1997 for both rowrnand broadcast sown methods. This indicates that the coefficient of variation and plot sizesrnare inversely proportional. Nevertheless, the coefficients of variation for 1996 sown byrnbroadcast method have higher values than the other year of the same method. This showsrnthat the coefficient of variation was influenced not only by soil variability and plotrnorientation but also by crop geometry resulting from the sowing method. The test donernfor identification and adaptation of different varieties on different location shows thatrnthere is no average effect due to variety and hence all varieties have the same effect onrnthe response variable. This implies that all varieties are contributing the same averagerneffects for the yield obtained.rn9rnHowever, the test statistics for location shows that the coefficients for Bako, Bekoji,rnDebre Zeit, Ginchi, Hollota, Kulumsa, and Sinana are significantly different from thernremaining locations. This significance indicates that the average productivity potential ofrnthese locations is higher than that of the remaining locations. In general, the contributionrnof each location to yield is not the same unlike that of variety. Hence, the average effectrnof location on the response variable is not the same.