ࡱ>   bbjbj #Nu&(((8`4(~P"c_-`Ly`(g~i~i~i~i~i~i~$Ҁv~`fYc_`f`f~9~sss`fg~s`fg~ssN{{&}]*j3}0S~~0~c}0oV`}}`ubNsc d```~~dqP```~`f`f`f`f````````` : Topic 13. Analysis of Covariance (ANCOVA, ST&D Chapter 17) 13. 1. Introduction The ANCOVA is useful for improving the precision of an experiment. Suppose that Y is linearly related to X. Furthermore, suppose that the experimenter cannot control X but can observe it along with Y. The variable X is called a covariate or concomitant variable. The analysis of covariance involves adjusting the observed response variable for the effect of the covariate. If such an adjustment is not performed, the covariate could inflate the error mean square and make true differences in the response due to treatments harder to detect. The concept is similar to the use of blocks to reduce the experimental error. However, when the blocking variable is a continuous variable, the delimitation of the blocks can be very subjective. The procedure is a combination of analysis of variance and regression analysis. We will review briefly some concepts of regression analysis to facilitate the discussion of the analysis of covariance. 13. 2. Review of Regression concepts. The equation of a straight line is Y= a + bX, Y is the dependent variable and X is the independent variable. a is called the intercept. The coefficient b is the slope Any point (X, Y) on this line has an X coordinate, or abscissa, and a Y coordinate, or ordinate. Body weight, X Food consumption, Y 4.6 87.1 5.1 93.1 4.8 89.8 4.4 91.4 5.9 99.5 4.7 92.1 5.1 95.5 5.2 99.3 4.9 93.4 5.1 94.4 Yestimated = 55.26 + 7.69* X_  13. 2. 1. The principle of least squares: the line that best fits the data. For each point we find its vertical distance from the straight line, square this distance, and then add together all the squared distances. The best-fitting line is the one that minimizes the sum of squared vertical deviations. 13. 2. 2. Residuals: The distance from a point to the straight line is a residual The residuals represent the behavior of Y that the independent variables dont account forthe error in the model. 13. 2. 3. Formulas to calculate a and b: The equations for the intercept a and the slope b the minimize the SS of the residuals are: 13. 2. 4. Covariance: S(XY) is called the corrected sum of cross products. Dividing S(XY) by n-1 produces a statistic called the sample covariance. If high values of Y are associated with high of X the covariance will be positive. If high values of Y are associated with low values of X the covariance will be negative. If there is no association the covariance will be close to zero. 13. 2. 5. Using SAS for regression analysis: PROC REG and PROC GLM. data STp254_r; input x y @@; cards; 4.6 87.1 5.1 93.1 4.8 89.8 4.4 91.4 5.9 99.5 4.7 92.1 5.1 95.5 5.2 99.3 4.9 93.4 5.1 94.4 ; proc glm; model y= x; run; quit; Note that the independent variable is not included in a CLASS statement Output: Sum of Mean Source DF Squares Square F Value Pr > F Model 1 90.84 90.84 16.23 0.0038 Error 8 44.77 5.60 Corr. Total 9 135.60 F: tests if the model as a whole accounts for a significant proportion of variation in Y. R-Square C.V. Root MSE Y Mean 0.670 2.528 2.366 93.560 R-Square: measures how much variation in Y the model can account for. T for H0: Pr > |T| Std Error of Parameter Estimate Parameter=0 Estimate INTERCEPT (a) 55.26 5.80 0.0004 9.5349 X (=b) 7.69 4.03 0.0038 1.9087 Estimates: Calculates the INTERCEPT (a= 55.26) and the slope (b= 7.69) and test if they are significantly different from 0. 13. 2. 6. ANOVA of the adjusted Ys The MSerror (44.77) from the previous analysis represents the variation in Y (food consumption) that would have been obtained if all the animals used in the experiment had had the same initial (X) body weight. In the following table each Y value is adjusted using the regression equation to a common X. Y= Y + b( EMBED Equation.2 - X) same as Y= Y - b(X - EMBED Equation.2 ) Any value of X can be used to adjust the Ys but the mean of the X ( EMBED Equation.2 = 4.98) value is used as a representative value: X YAdjusted Y= Y + b( EMBED Equation.2 - X)4.687.190.025.193.192.184.889.891.184.491.495.865.999.592.434.792.194.255.195.594.585.299.397.614.993.494.025.194.493.48  EMBED Equation.2 = 4.98SSY 135.60444.77The first adjusted value. 90.02, is the expected food consumption expected for this animal if its initial body weight would have been 4.98. Note that the SS of the Ys is similar to the Total SS of the previous ANOVA and that the SS of the adjusted Ys is similar to the SSerror. The SSerror is the variation in food consumption that we would have found if all the animals used in the experiment had had the same weight (assuming that b was estimated without error). Note the large reduction in the variation of the Ys that is obtained when the variation due to the regression is eliminated. 13. 3. ANCOVA example The goal of this experiment with oysters is to determine if exposure to water heated artificially affects growth if the position in the water column (surface or bottom) affects growth Four bags (e.u.) with ten oysters in each bag are randomly placed at each of 5 locations in the cooling water canal of a power-generating plant. TRT1: cool-bottom, TRT2: cool-surface, TRT3: hot-bottom, TRT4: hot-surface, TRT5: control mid-depth and mid-temperature. Each bag is weighted at the beginning and at the end of the experiment. data oyster; input trt rep initial final; cards; 1 1 27.2 32.6 1 2 32.0 36.6 1 3 33.0 37.7 1 4 26.8 31.0 2 1 28.6 33.8 2 2 26.8 31.7 2 3 26.5 30.7 2 4 26.8 30.4 3 1 28.6 35.2 3 2 22.4 29.1 3 3 23.2 28.9 3 4 24.4 30.2 4 1 29.3 35.0 4 2 21.8 27.0 4 3 30.3 36.4 4 4 24.3 30.5 5 1 20.4 24.6 5 2 19.6 23.4 5 3 25.1 30.3 5 4 18.1 21.8 ; proc GLM; Title 'One-way Anova'; class trt; model final= trt; proc GLM; Title 'ANCOVA'; class trt; model final= trt initial; run; quit; The CLASS statement specifies that TRT is a classification variable. The variable initial is the covariate. Note that initial is a continuous variable and is NOT included in the CLASS statement. This is similar to the regression example in 13. 2. 4. A simple regression of Final versus Initial showed a large significant regression (P<0.01, R2= 0.95). Note that the slopes of the regressions within the different treatments are similar. Examples of final vs. initial regressions for treatments 1 and 3 are shown below.  TRT1: Y= 5.24 +0.98X TRT3: Y= 4.82 +1.06X SAS Output: One-way ANOVA Dependent Variable: FINAL Source DF SS MS F Value Pr > F TRT(= Model) 4 198.407 49.60 4.64 0.0122 Error 15 160.262 10.68 Correc. Total 19 358.669 R-Square C.V. Root MSE FINAL Mean 0.553175 10.597 3.2687 30.845 The simple one-way ANOVA discovers treatment differences (P= 0.0122) in final weight even when the initial weights are not considered ANCOVA Dependent Variable: FINAL Source DF SS MS F Value Pr > F Model 5 354.45 70.89 235.05 0.0001 Error 14 4.22 0.30 Corrected Total 19 358.67 R-Square C.V. Root MSE FINAL Mean 0.988228 1.780438 0.5492 30.845 Source DF Type I SS Mean Square F Value Pr > F TRT 4 198.40700 49.60175 164.47 0.0001 INITIAL 1 156.04018 156.04018 517.38 0.0001 Source DF Type III SS Mean Square F Value Pr > F TRT 4 12.08936 3.02234 10.02 0.0005 INITIAL 1 156.04018 156.04018 517.38 0.0001 The Type I SS for TRT is the unadjusted treatment SS and is the same as the one found in the one-way ANOVA. The Type III SS (12.089) is the adjusted treatment SS and enable us to test the treatment effects adjusted for all other factors included in the model. Type III SS will produce the appropriate results for the ANCOVA. Though this SS is smaller than the unadjusted TRT SS (198.407) the reduction in the error term is even larger (from 10.684 to 0.30159). This allows an increase in the F statistic of the test from 4.642 in the simple one-way ANOVA to 10.02 in the ANCOVA. The power of the test for treatment differences increases when the covariate is included because most of the error in the simple ANOVA is due to variation in INITIAL values. Finally, the INITIAL SS is used to test the significance of the regression between INITIAL and FINAL once the SS of INITIAL was adjusted for TRT. 13. 3. 1. Graphic interpretation of the ANCOVA example            The red circles represent the data from the four bags of treatment 2. The blue squares represent the data from treatment 3. The mean final weight of treatment 3 (30.85) is slightly lower than the mean of treatment 2 (31.65). For each treatment, variation in X is seen to contribute to variation in Y. The distance between the initial weight averages of the oysters assigned to each treatment can contribute greatly to the difference between the final average weights. Treatment 3 started with an average initial weight of 24.65 and treatment 2 with an average weight of 27.175. If the treatments means had been observed from some common average X, say Xo, then they would be comparable. Thus the need for adjusting treatment means is apparent. This Xo can be thought as a common mean obtained after moving the values of treatment 3 upwards along the regression line and the values of treatment 2 downward along the regression line. The adjusted mean of treatment 3 is significantly larger than that of treatment 2. 13. 3. 2. Least squares means or adjusted means MEANS statement: unadjusted treatment means of all continuous (non-CLASS) variables in the model. These means or their comparison are not strictly appropriate. To be comparable, treatment means should be adjusted to make them the best estimates of what they would have been if all treatment independent means had been the same. These adjusted means can be calculated in SAS by using the LSMEANS (least-squares means) statement. This statement is the same one used to obtain adjusted means for the unbalanced two-way classification. These adjusted means are obtained using the following formula: The statement: lsmeans trt /stderr pdiff adjust=tukey; will print lsmeans followed by their standard errors and significance probabilities for all pairwise tests of treatment differences. Note the large changes between unadjusted and adjusted treatments means: TRTUnadjusted MeansAdjusted LS MeansCalculation134.47530.15334.475 - 1.08318 (29.75 - 25.76)231.65030.11731.650 - 1.08318 (27.18 - 25.76)330.85032.05230.850 - 1.08318 (24.65 - 25.76)432.22531.50432.225 - 1.08318 (26.43 - 25.76)525.02530.39825.025 - 1.08318 (20.80 - 25.76)These changes are due to the large trt differences for the variable INITIAL. Some treatments, particularly TRT 5, received smaller oysters than other trt. The coefficient (=1.08318 = weighted average of the regression coefficients of FINAL on INITIAL, for each of the 5 treatment groups. To obtain this coefficient: model final = trt initial/ solution; 13. 3. 3. Contrasts The adjusted treatment means from the analysis of covariance can be analyzed further with four orthogonal contrasts implemented by the following CONTRAST statements: proc GLM; class trt; model final= trt initial; contrast 'control vs. Treatment' TRT -1 -1 -1 -1 4; contrast 'bottom vs. top' TRT -1 1 -1 1 0; contrast 'cool vs, hot' TRT -1 -1 1 1 0; contrast 'interactions depth*temp' TRT 1 -1 -1 1 0; Output: Contrast DF Contrast SS MS F Value Pr > F control vs. Trt. 1 0.520 0.520 1.72 0.2103 bottom vs. top 1 0.339 0.339 1.12 0.3071 cool vs. hot 1 8.591 8.591 28.49 0.0001 interac. depth*T 1 0.229 0.229 0.76 0.3979 9.679 Source DF Type III SS MS F Value Pr > F TRT 4 12.089 3.022 10.02 0.0005 The only significant difference is cool vs. hot. These contrasts are not orthogonal to the covariable; hence, their sums of squares do not add to the adjusted treatment SS. If the covariable is not included in the model, the same contrast statements produce completely different results! Contrast DF Contrast SS MS F Value Pr > F control vs. Trt 1 169.36 169.36 15.85 0.0012 bottom vs. top 1 2.10 2.10 0.20 0.6637 cool vs. hot 1 9.30 9.30 0.87 0.3655 interac. depth*T 1 17.64 17.64 1.65 0.2183 The significance of the control vs. treatment contrast is due to the lower initial weight of the oysters placed in the control bags. In this case the contrast SS will add to the TRT SS (198.407). The contrast statement will partition the TRT SS resulting for the specified model. ANCOVA of Y-X: Oyster Example using Final Initial weight TRT:5 Levels:1 2 3 4 5 Number of observations in data set = 20 ANOVA(no covariable) Dependent Variable: FINAL(weight) Source DF SS MS F Value Pr > F TRT 4 198.41 49.60 4.64 0.0122 Error 15 160.26 10.68 Corrected Total 19 358.67 Dependent Variable: DIF (Final - Initial) Source DF SS MS F Value Pr > F TRT 4 11.98 3.00 8.74 0.0008 Error 15 5.14 0.34 Corrected Total 19 17.13 ANCOVA (using Initial as covariable) Dependent Variable: FINAL(weight) Source DF SS MS F Value Pr > F Model 5 354.45 70.89 235.05 0.0001 Error 14 4.22 0.30 Corrected Total 19 358.67 Source DF Type III SS MS F Value Pr > F TRT 4 12.09 3.02 10.02 0.0005 INITIAL 1 156.04 156.04 517.38 0.0001 Dependent Variable: DIF(Final - Initial) Source DF SS MS F Value Pr > F Model 5 12.90 2.58 8.56 0.0007 Error 14 4.22 0.30 Corrected Total 19 17.13 Source DF Type III SS MS F Value Pr > F TRT 4 12.09 3.02 10.02 0.0005 INITIAL 1 0.92 0.92 3.05 0.1026 Comparison between ANCOVA and ANOVA of ratios Effect of stress on the presence of enzyme A in the liver. The researcher measured the total activity of enzyme A from liver homogenates of 10 control and 10 shocked animals and the total amount of N to correct by the total protein present in the liver. A/N= activity of enzyme per unit of protein. Control animalsShocked animalsNAA/NNAA/N847690.412210888.52838133.998158161.21667243.41155850.0986465.3866575.51055350.0694058.0842832.8866575.5723143.01028280.3802834.31129484.1842832.7986566.31054946.17476102.7ANOVA for the variable A/N. Dependent Variable: A/N Source DF SS MS F Value Pr > F Model = TRT 1 3650.40 3650.40 3.66 0.0719 Error 18 17970.72 998.37 Correc. Total 19 21621.12 The ANOVA indicates that there are no significant differences between treatments. ANCOVA for the variable A using N as covariable. Dependent Variable: A Source DF SS MS F Value Pr > F Model 2 8079.39 4039.70 6.31 0.0089 Error 17 10882.41 640.14 Correc. Total 19 18961.80 Source DF Type III SS MS F Value Pr > F TRT 1 5108.81 5108.81 7.98 0.0117 N 1 2162.59 2162.59 3.38 0.0836 The ANCOVA shows significant differences between treatments. The use of ANOVA to analyze ratios Z = Y/X is not that correct. Both X and Y have random variation Variation in the numerator (Y) affect Z in a lineal way but variation in the denominator affect Z in a hyperbolic way The magnitude of the error of Z depends not only on the error of X but also on the absolute value of X. 13. 4 ANCOVA model The ANOVA model for a CRD is  The regression model is: The ANCOVA model for a CRD is: The linear additive model for any given design is that for the ANOVA plus an additional term for the concomitant or independent variable. Y generally denotes the dependent variable, whereas the variable used in the control of error and adjustment of means, is denoted by X. The rearranged formula  ANCOVA is a regular ANOVA of values that have been adjusted for regression on an independent variable. 13. 5 Assumptions of the ANCOVA The Xs are fixed, measured without error, and independent of treatments. This means that the measurement error is trivial relative to the observed variation, and that the treatments itself will not affect the X values. The regression of Y on X after removal of the treatment differences is linear and independent of treatments. This means that the regression is assumed to be approximately linear within the range of X values, and that the slopes of the regressions within the treatments are not significantly different. The residuals are normally and independently distributed with zero mean and a common variance. These are the normal assumptions for the validity of the F tests. 13. 5. 1. Independence of X values from the treatments If the covariable is measured before the experiment, like in the previous oyster example (13.3) the independence of the treatments and the concomitant variable is always satisfied. If the concomitant variable is measured after the experiment the independence of the covariable and the treatments should be tested. An analysis of variance of the covariable using the treatments as CLASS variable is appropriate to test this hypothesis. The null hypothesis is that there are no significant differences among treatments for the covariable. We expect to find no significant differences in order to be able to perform a standard covariance analysis. The following statements are included only as an example, because the test is not required in the oyster example. proc glm; title 'Test for independence of treatments and covariable'; class trt; model initial= trt; run; quit; Dependent Variable: INITIAL Source DF SS MS F Value Pr > F TRT (Model) 4 176.793 44.198 4.98 0.0093 Error 15 132.995 8.866 R-Square Coeff Var Root MSE Initial Mean 0.570690 11.55916 2.977639 25.76 In this case the differences in initial weight are highly significant. The selected covariable will most likely have an effect in the final results. Covariance can be used where the X values are affected by the treatments but it must be interpreted with caution. Values in red will be used later for a correction in the calculation of the efficiency of the ANCOVA relative to the ANOVA The Mean of the covariable will be used to obtain the corrected Y values 13. 5. 2. Test for heterogeneity of slopes. The adjustment of the Y values using a single ( for all treatments is based on the assumption of a constant regression relationship among groups The test for heterogeneity of slopes tests the validity of this assumption The null hypothesis is Ho: (1 = (2 = ... = (i Regression relationships that differ among treatments reflect an interaction between the treatment groups and the covariates. The GLM procedure analyzes this phenomenon as an interaction. The expression X*TRT produces the appropriate statistics for estimating different regressions of Y on X for the different classes specified by TRT. For the previous example (Topic 13.3) the appropriate statements are proc glm; title Test for heterogeneity of slopes; class trt; model final= trt initial trt*initial; run; quit; Output: Dependent Variable: FINAL Source DF SS MS F Value Pr > F Model 9 355.84 39.54 139.51 0.0001 Error 10 2.83 0.28 Corrected Total 19 358.67 Source DF Type I SS MS F Value Pr > F TRT 4 198.407 49.60 175.02 0.0001 INITIAL 1 156.040 156.04 550.60 0.0001 INITIAL*TRT 4 1.388 0.35 1.22 0.3602 Source DF Type III SS MS F Value Pr > F TRT 4 1.697 0.424 1.50 0.2752 INITIAL 1 68.529 68.529 241.81 0.0001 INITIAL*TRT 4 1.388 0.347 1.22 0.3602 The last row INITIAL*TRT is an additional SS due to different regression coefficients for the groups specified by TRT. If INITIAL*TRT P>0.05 ( dont reject hypothesis of slope homogeneity. 13. 5. 3. Analysis of residuals. Similar statements to those used in previous analysis can be included to check the normality and homogeneity of variance of the residuals: proc GLM; class trt; model final= trt initial; output out=check p=predi r=resi; proc univariate data=check normal; var resi; proc plot; plot resi*predi=trt; run; quit; 13. 6. Increase in precision due to covariance (ST&D 17.6) To test the effectiveness of covariance to control error we compare the variance of treatment mean with and without the covariance adjustment. The ANOVA on the unadjusted Ys has error MS 10.68417 with 15 df The ANCOVA on the adjusted Ys has error MS 0.30159 with 14 df This last value must be adjusted upward to allow for sampling error in the regression coefficient. The adjustment involves the TRT SS (176.793) and the error SS (132.995) from an ANOVA on X. The effective error MS after adjustment for X is given by An estimate of the relative precision is : MSError Unadjusted Y / MSError Adjusted Y = 10.68417 / 0.402= 26.6, This indicates that each replication in the ANCOVA is as effective as 26.6 replications in the ANOVA. Assumptions Data Adjust; Input Block Variety X Y @@; Z= Y - 8.39163498 * (X - 4.98); * Z is adjusted Y; Cards; 1 1 4.6 87.1 2 1 4.7 92.1 1 2 5.1 93.1 2 2 5.1 95.5 1 3 4.8 89.8 2 3 5.2 99.3 1 4 4.4 91.4 2 4 4.9 93.4 1 5 5.9 99.5 2 5 5.1 94.4; Proc GLM Data = Adjust; * Model to test ANOVAS; Class Block Variety; Model X Y = Block Variety; Proc GLM Data = Adjust;* The ANCOVA; Class Block Variety; Model Y = Block Variety X / solution; output out=check p=predi r=resi; lsmeans Variety; Proc univariate data=check normal; var resi; * Test of normality of residuals; Proc GLM Data = Adjust; * The ANOVA of Z; Class Block Variety; Model Z = Block Variety; * Produces similar P ANCOVA; Output Out = ZZ p = PZ r = RZ; means var; * Produces = result as LSMEANS in ANCOVA; Proc GLM Data = ZZ; Class Variety; Model Z = Variety; means Variety / hovtest= Levene; * Levenes test adjusted Z; Proc GLM Data = ZZ; Class Block Variety; Model Z = Block Variety PZ*PZ; *Tukey test non-additivity Z; Run; quit; ANOVA Dependent Variable: X Source DF SS MS F Value Pr > F Model 5 1.01 0.20 1.54 0.3492 Error 4 0.53 0.13 Corrected Total 9 1.54 R-Square Coeff Var Root MSE X Mean 0.657552 7.281713 0.362629 4.98 solution Standard Parameter Estimate Error t Value Pr > |t| Intercept 52.00817490 B 19.46736895 2.67 0.0756 Block 1 -2.42433460 B 1.61546874 -1.50 0.2304 Block 2 0.00000000 B . . . Variety 1 -0.21711027 B 3.92035053 -0.06 0.9593 Variety 2 0.70665399 B 2.90598043 0.24 0.8236 Variety 3 1.79581749 B 3.09073299 0.58 0.6020 Variety 4 2.58288973 B 3.92035053 0.66 0.5571 Variety 5 0.00000000 B . . . X 8.39163498 3.50857389 2.39 0.0966 13. 8. Uses of ANCOVA (ST&D p 429) 1. To control error and increase precision 2. To adjust treatment means of the dependent variable for differences in sets of values of corresponding independent variables 3. To assist in the interpretation of data, especially with regard to the nature of treatment effects 4. To estimate missing data 13. 8. 1. Error control. Experimental design, a covariable, or both can be used to reduce MS error Covariance can be used to reduce error when variation in the dependent variable Y is partly attributable to variation in the independent variable X. Removes, by regression, certain recognized effects that cannot be or have not been controlled effectively by experimental design. Example: cattle-feeding experiment to compare the effects of several rations on gain in weight, using initial weight as covariable. 13. 8. 3. To adjust treatment means When variation in Y is partly attributable to variation in X, variation in treatment  EMBED Equation.2  must also be affected by variation among  EMBED Equation.2 . To be comparable, treatment  EMBED Equation.2  should be adjusted to make them the best estimates of what they would have been if all treatment  EMBED Equation.2  had been =. Example1: canning peas yield increases rapidly with increase in maturity. Maturity can be used as a covariate. A comparison of yields adjusted for maturity differences would be more meaningful than a comparison among unadjusted yields. Example2: In animal feeding experiments, differences among unadjusted treatment means may be due to differences in the nutritive value of the rations, to differences in the amounts consumed, or to both. If differences among mean gains in weight for the different rations are adjusted to a common food intake, the adjusted means indicate whether or not the rations differ in nutritive value. 13. 7. 3. Interpretation of data. Covariance analysis often aids the experimenter in understanding the principles underlying the results of an investigation. If the independent variable is influenced by the treatments, the interpretation of the data is changed. Adjustment removes part of the treatment effects when means of the independent variable are affected by treatments. The adjusted treatment means estimate values expected when treatment means for X are =. Example 1: In a fertilizer trial on sugar beets, the treatments may cause differences in stand. When stand, the independent variable, is influenced by treatments, the analysis of yield adjusted for stand differences removes part of the treatment effect. An analysis of covariance can still supply useful information. Total yield= average weight per beet x No. of beet (stand). If stand is influenced by treatments, the ANCOVA of yield adjusted for stand differences would indicate treatments affects in weight per beet. Example 2: An adjustment in proportion to the number of plants is sometimes practiced. The use of ratios is not recommended, because it usually results in an over-correction for the plots with smallest stand since yields are rarely proportional to the number of plants per plot. The analysis of covariance provides a more satisfactory and appropriate method of adjusting the experimental data. In situations where real differences among treatments for the independent variable do occur but are not the direct effect of the treatments, adjustment is warranted. For example, consider a variety trial for which seed of the various varieties or strains has been produced in different areas. Such seed may differ widely in germination, not because of inherent differences, but as a result of the environment in which it was grown. Differences in stand may occur even if planting rate is controlled. In this situation, the use of covariance for both error control and yield adjustment is warranted. 13. 7. 4. Estimation of missing data: The use of covariance to estimate missing plots results in a minimum residual sum of squares and an unbiased treatment sum of squares. The covariance procedure is simple to carry out though more difficult to describe than previous procedures which required little more than a formula. Interpretation of examples of ANCOVA 1) Effect of a fertilizer treatment on sugar beets. Variable: yield Covariable: number of plants. Treatment Treatment Treatment ANOVA significant significant NS ANCOVA significant NS significant 2) Effect of different cat foods on weight gain during the first year. Variable: gained weight Covariable: food intake. Treatment Treatment Treatment ANOVA significant significant NS ANCOVA significant NS significant 3) Effect of two storage protein genes on bread loaf volume (2x2 factorial) Variables: loaf volume Covariable: protein content Treatment Treatment Treatment ANOVA G1= S G2=S G1=NS G2=NS G1=S G2=S ANCOVA G1=NS G2=NS G1=S G2=S G1=S G2=S 4) Effect of hardness genes and environment in grain hardness Variables: grain hardness Covariable: grain weight Treatment Treatment ANOVA G=S E=S GxE=S G=NS E=S GxE=S ANCOVA G=S E=NS GxE=NS G=S E=NS GxE=S     PAGE  PAGE 17  EMBED Word.Picture.8   EMBED Word.Picture.8   EMBED Equation.3  Hot water COLD INTERMEDIATE HOT TOP BOTTOM 1 2 3 4 5  EMBED Word.Picture.8   EMBED Word.Picture.8  Y= final weight (X2, Y2) (X3, Y3) Adjusted difference Observed difference Xo X= initial weight TRT3 24.65 TRT2 27.175  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  +:<P) * 7 @ A D X  X l q { " G S T \ e s ײםײײדׇׇwh~h556B*CJph *h~h55CJ *h~h5CJ h~CJh~h~5B*CJphh~h55B*CJphh~h55B*CJph h56CJ h5CJ h55CJh~h5B* CJphh~h55B* CJph*<P[ r 6 $ S 1 ^ i t   !* !"$d&d-DM NP]"`gd)1 !x`gd~  & F !x  & F !x  !x`s t |     ' / 1 2 X ǹ馝|njc h{A5CJh{Ah5CJehr!h56CJH*ehrh56CJehrh)1h5CJ$jh)1h5CJUmHnHu h~6CJ h55CJ h56CJ *h~h55CJh~h55B*CJph h5CJh~h556B*CJph! 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