Number of grains per head of barley (NumGrains), with seeding rate (SeedRate). The seeding rates are 50, 75, 100, 125, 150, coded here as factors. SeedRate NumGrains 1 1 21.38440 2 1 20.85252 3 1 20.93921 4 1 21.03766 5 1 21.25097 6 2 19.83274 7 2 20.04814 8 2 20.31631 9 2 19.64763 10 2 20.30815 11 3 19.03917 12 3 19.31179 13 3 19.04618 14 3 19.27339 15 3 18.98670 16 4 18.22121 17 4 18.60824 18 4 18.82809 19 4 18.14340 20 4 18.20491 21 5 18.02074 22 5 17.65943 23 5 17.92460 24 5 17.52438 25 5 17.57863 > is.factor(barley$SeedRate) [1] FALSE > barley$SeedRate=factor(barley$SeedRate) > summary(lm(NumGrains ~ SeedRate,data=barley)) Call: lm(formula = NumGrains ~ SeedRate, data = barley) Residuals: Min 1Q Median 3Q Max -0.38296 -0.17996 -0.08213 0.18304 0.42692 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 21.0930 0.1088 193.930 < 2e-16 *** SeedRate2 -1.0624 0.1538 -6.907 1.04e-06 *** SeedRate3 -1.9615 0.1538 -12.752 4.61e-11 *** SeedRate4 -2.6918 0.1538 -17.500 1.36e-13 *** SeedRate5 -3.3514 0.1538 -21.788 2.09e-15 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.2432 on 20 degrees of freedom Multiple R-squared: 0.9674, Adjusted R-squared: 0.9608 F-statistic: 148.2 on 4 and 20 DF, p-value: 1.466e-14 > anova(lm(NumGrains ~ SeedRate,data=barley)) Analysis of Variance Table Response: NumGrains Df Sum Sq Mean Sq F value Pr(>F) SeedRate 4 35.057 8.764 148.17 1.466e-14 *** Residuals 20 1.183 0.059 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > options("contrasts") $contrasts unordered ordered "contr.treatment" "contr.poly" > options(contrasts = c("contr.sum","contr.poly")) > summary(lm(NumGrains ~ SeedRate,data=barley)) Call: lm(formula = NumGrains ~ SeedRate, data = barley) Residuals: Min 1Q Median 3Q Max -0.38296 -0.17996 -0.08213 0.18304 0.42692 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 19.27954 0.04864 396.360 < 2e-16 *** SeedRate1 1.81341 0.09728 18.641 4.12e-14 *** SeedRate2 0.75105 0.09728 7.720 2.01e-07 *** SeedRate3 -0.14810 0.09728 -1.522 0.144 SeedRate4 -0.87837 0.09728 -9.029 1.71e-08 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.2432 on 20 degrees of freedom Multiple R-squared: 0.9674, Adjusted R-squared: 0.9608 F-statistic: 148.2 on 4 and 20 DF, p-value: 1.466e-14 > mean(barley$NumGrains) [1] 19.27954 > barley$SeedRate = as.ordered(barley$SeedRate) > summary(lm(NumGrains ~ SeedRate,data=barley)) Call: lm(formula = NumGrains ~ SeedRate, data = barley) Residuals: Min 1Q Median 3Q Max -0.38296 -0.17996 -0.08213 0.18304 0.42692 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 19.27954 0.04864 396.360 < 2e-16 *** SeedRate.L -2.63488 0.10877 -24.225 2.70e-16 *** SeedRate.Q 0.26041 0.10877 2.394 0.0266 * SeedRate.C -0.02927 0.10877 -0.269 0.7906 SeedRate^4 -0.01242 0.10877 -0.114 0.9102 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.2432 on 20 degrees of freedom Multiple R-squared: 0.9674, Adjusted R-squared: 0.9608 F-statistic: 148.2 on 4 and 20 DF, p-value: 1.466e-14 > contrasts(barley$SeedRate) .L .Q .C ^4 1 -0.6324555 0.5345225 -3.162278e-01 0.1195229 2 -0.3162278 -0.2672612 6.324555e-01 -0.4780914 3 0.0000000 -0.5345225 -4.095972e-16 0.7171372 4 0.3162278 -0.2672612 -6.324555e-01 -0.4780914 5 0.6324555 0.5345225 3.162278e-01 0.1195229 > aov(NumGrains ~ SeedRate,data=barley,contrasts=contrasts) Call: aov(formula = NumGrains ~ SeedRate, data = barley, contrasts = contrasts) Terms: SeedRate Residuals Sum of Squares 35.05703 1.18300 Deg. of Freedom 4 20 Residual standard error: 0.2432074 Estimated effects are balanced > summary(.Last.value,split=list(SeedRate=list(L=1,Q=2,C=3,QQ=4)) + ) Df Sum Sq Mean Sq F value Pr(>F) SeedRate 4 35.057 8.764 148.1705 1.466e-14 *** SeedRate: L 1 34.713 34.713 586.8641 2.704e-16 *** SeedRate: Q 1 0.339 0.339 5.7323 0.02657 * SeedRate: C 1 0.004 0.004 0.0724 0.79062 SeedRate: QQ 1 0.001 0.001 0.0130 0.91024 Residuals 20 1.183 0.059 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1