Run ANCOVA correctly in R software

Briefly, covariance analysis (ANCOVA) is expressed in the R language as y ~ x +/* y0, where the variable is y, the covariate is y0, the independent variable is x. Then, go ahead code test.

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data.of.Wada2013.practice <- read.csv("~data of Wada2013 practice.csv")

View(data.of.Wada2013.practice)

eco<-data.of.Wada2013.practice

eco

\ site d13C d15N

1. 1 -28.40 4.05

2. 1 -26.90 6.70

3. 1 -25.90 7.30

4. 1 -25.85 7.85

5. 1 -25.60 9.90

6. 1 -26.35 10.30

7. 1 -26.25 10.70

8. 1 -26.55 12.20

9. 1 -26.60 12.65

10. 1 -26.50 13.25

11. 1 -26.40 13.70

12. 1 -24.90 14.10

13. 2 -24.00 9.02

14. 2 -23.50 11.80

15. 2 -22.71 15.78

16. 2 -22.10 15.10

17. 2 -20.50 14.00

18. 2 -19.75 17.30

19. 2 -20.35 17.95

20. 3 -23.80 3.95

21. 3 -23.50 4.80

22. 3 -23.80 5.20

23. 3 -23.70 6.10

24. 3 -22.80 6.95

25. 3 -21.95 7.10

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attach(eco)

sitef<-factor(eco$site)
#Step1: ANCOVA with InterAction
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mod.IA<-aov(d15N~d13C*sitef,data=eco,  contrasts=list(sitef=contr.sum))
N.B., aov() function, “contr.sum” for factor “sitef”(not numerical “site”)
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summary(mod.IA)   # this summary is wroooong, it's for type I , not type III, don't do this!
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library(car)

Anova(mod.IA,type=3) # this is riiiiight. Anova() function, not anova

Anova Table (Type III tests)

Response: d15N

\ Sum Sq Df F value Pr(>F)

(Intercept) 70.098 1 13.0456 0.001858 **

d13C 43.563 1 8.1073 0.010302 *

sitef 3.341 2 0.3109 0.736451

d13C:sitef 1.771 2 0.1648 0.849262

Residuals 102.093 19

Signif. codes: 0 * ,0.001 ,0.01 * ,0.05 . ,0.1 , 1

covariate d13C is significantly differ with d15N ( p = 0.001858),

and interation of d13C:sitef is not significantly differ ( p = 0.849262 ),

——-> then we can proceed ANCOVA next.


Step2: then ANCOVA NonInterAction, for equal slope, type III
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 mod.NIA<-aov(d15N~d13C+sitef,data=eco,contrasts = list(sitef=contr.sum))

Anova(mod.NIA, type=3)

Anova Table (Type III tests)

Response: d15N

\ Sum Sq Df F value Pr(>F)

(Intercept) 117.10 1 23.677 8.232e-05 ***

d13C 74.32 1 15.027 0.0008727 ***

sitef 258.75 2 26.158 1.989e-06 ***

Residuals 103.86 21

Signif. codes: 0 *, 0.001 , 0.01 *, 0.05 ., 0.1 , 1

p value of 8.232e-05 *** means intercept is significantly not equal to 0.

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summary.lm(mod.NIA)

Call:

aov(formula = d15N ~ d13C + sitef, data = eco, contrasts = list(sitef = contr.sum))

Residuals:

Min 1Q Median 3Q Max

-3.6695 -1.5658 0.1179 1.1474 3.5577

Coefficients:

\ Estimate Std. Error t value Pr(>|t|)

(Intercept) 49.51511 0.1760 4.866 8.23e-05 ***

d13C 1.6545 0.4268 3.876 0.000873 ***

sitef1 4.3049 1.2335 3.490 0.002183 **

sitef2 1.0468 1.0774 0.972 0.342273

Signif. codes: 0 *, 0.001 , 0.01 *, 0.05 ., 0.1 , 1

Residual standard error: 2.224 on 21 degrees of freedom

Multiple R-squared: 0.7556, Adjusted R-squared: 0.7207

F-statistic: 21.65 on 3 and 21 DF, p-value: 1.252e-06

Based on above, we could get the equation with equal slope and common intercept of 3 sites which suitable with Wada_2013 as: d15N=1.6545[±0.4268]*d13C+49.51511[±0.1760 ]+[ecosystem specific sites constant] ( p = 0.000873). Step3: then ANCOVA NonInterAction with treatment contrasts, or type I.
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mod.NIA.tc<-aov(d15N~d13C+sitef,data=eco,contrasts = list(sitef=contr.treatment))

summary.lm(mod.<font color=red>NIA</font>.tc) #this step is riiiight,for graphics with type I is what we need.

Call:

aov(formula = d15N ~ d13C + sitef, data = eco, contrasts = list(sitef = contr.treatment))

Residuals:

Min 1Q Median 3Q Max

-3.6695 -1.5658 0.1179 1.1474 3.5577

Coefficients:

\ Estimate Std. Error t value Pr(>|t|)

(Intercept) 53.8200 11.2645 4.778 0.000101 ***

d13C 1.6545 0.4268 3.876 0.000873 ***

sitef2 -3.2581 2.1947 -1.485 0.152529

sitef3 -9.6567 1.7256 -5.596 1.49e-05 ***

Signif. codes: 0 *, 0.001 , 0.01 *, 0.05 ., 0.1 , 1

Residual standard error: 2.224 on 21 degrees of freedom

Multiple R-squared: 0.7556, Adjusted R-squared: 0.7207

F-statistic: 21.65 on 3 and 21 DF, p-value: 1.252e-06

Based on above, we could get the equation for 3 sites respectively with equal slope and specific intercepts which suitable with Wada_2013 as:

site1: d15N=1.6545d13C+53.8200 ;

site2: d15N=1.6545
d13C+53.8200-3.2581 ;

site1: d15N=1.6545*d13C+53.8200-9.6567 .
Step4: Graphics
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dat1<-subset(eco,sitef == "1")                  

dat2<-subset(eco,sitef == "2")

dat3<-subset(eco,sitef=="3")

reg1<-lm(d15N~d13C,data=dat1)

reg2<-lm(d15N~d13C,data=dat2)

reg3<-lm(d15N~d13C,data=dat3)

plot(d15N~d13C,type="n") # type="n" means plot with axes,without any plots.

points(dat1$d13C,dat1$d15N,pch=1)

points(dat2$d13C,dat2$d15N,pch=2)

points(dat3$d13C,dat3$d15N,pch=3)

abline(reg1,lty=1)

abline(reg2,lty=2)

abline(reg3,lty=3)

legend("topleft",c("L.Baikal","L.Biwa","Mongolian grassland"),lty=c(1,2,3),pch=c(1,2,3))
Above procedures output a graph with unequal slopes which we also need.

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summod.NIA.tc<-summary.lm(mod.NIA.tc)       # type I

coeffs.NIA.tc<-coef(summod.NIA.tc)

coeffs.NIA.tc

\ Estimate Std. Error t value Pr(>|t|)

(Intercept) 53.820034 11.2645326 4.777831 1.014464e-04

d13C 1.654461 0.4268016 3.876416 8.726695e-04

sitef2 -3.258098 2.1947260 -1.484512 1.525294e-01

sitef3 -9.656707 1.7255811 -5.596206 1.488283e-05

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I1<-coeffs.NIA.tc[1,1]

I2<-coeffs.NIA.tc[3,1]+I1

I3<-coeffs.NIA.tc[4,1]+I1

solpeAll<-coeffs.NIA.tc[2,1]

plot(d15N~d13C,type="n")

points(dat1$d13C,dat1$d15N,pch=1)

points(dat2$d13C,dat2$d15N,pch=2)

points(dat3$d13C,dat3$d15N,pch=3)

abline(I1,slopeAll,lty=1)

abline(I2,slopeAll,lty=2)

abline(I3,slopeAll,lty=3)
Aboveall,we could clearly understand why we can not use summary.lm(mod.NIA) for graphics.
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legend("topleft",c("L.Baikal","L.Biwa","Mongolian grassland"),lty=c(1,2,3),pch=c(1,2,3))

OMG, so long so tired. Anyway, we have correctly achieved what we need, that’s enough ^_^



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