example2 {agriTutorial} | R Documentation |

## Example 2: Lack-of-fit and marginality for a single quantitative treatment factor

### Description

Petersen (1994, p. 125) describes an experiment conducted to assess the effects of five different quantities of N-fertiliser (0, 35, 70, 105 and 140 kg N/ha) on root dry matter yield of sugar beet (t/ha) with three complete replications laid out in three randomized complete blocks. One objective of this experiment was to determine the amount of fertilizer for maximizing yield.

### Details

The first stage of the analysis is the calculation of raw polynomial powers of N using the poly() function. The N rates are re-scaled by division by 100 to improve numerical stability.

The second stage fits a full polynomial analysis of variance based on polynomial contrasts which are fitted in sequence from the lowest to the highest. This is equivalent to the analysis shown in Tables 4 and 5 of Piepho and Edmondson (2018) except that a complete partition into single degree of freedom polynomial contrasts is shown here compared with the pooled 'lack of fit' term shown in Tables 4 and 5.

The third stage fits a quadratic regression model with linear and quadratic terms only. This model provides the model coefficients, standard errors and the confidence intervals shown in Table 6 of Piepho and Edmondson (2018). A set of diagnostic plots are fitted for the fitted quadratic regression model to check the validity of the model assumptions.

Finally, a smoothed quadratic graph of the yield versus the N rate is plotted to show the goodness of fit of the quadratic regression model. This plot corresponds to plot Fig 3 in Piepho and Edmondson (2018).

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### References

Petersen, R.G. (1994). Agricultural field experiments. Design and analysis. New York: Marcel Dekker.

Piepho, H. P, and Edmondson. R. N. (2018). A tutorial on the statistical analysis of factorial experiments with qualitative and quantitative treatment factor levels. Journal of Agronomy and Crop Science. DOI: 10.1111/jac.12267. View

### Examples

```
## *************************************************************************************
## How to run the code
## *************************************************************************************
## Either type example("example2") to run ALL the examples succesively
## or copy and paste examples sucessively, as required
## *************************************************************************************
## Options and required packages
## *************************************************************************************
options(contrasts = c('contr.treatment', 'contr.poly'))
## ggplot2 MUST be installed
require(ggplot2)
## *************************************************************************************
## Polynomial analysis and graphical plots of factorial treatment effects
## *************************************************************************************
N = poly((beet$nrate/100), degree = 4, raw = TRUE)
colnames(N) = c("Linear_N", "Quadratic_N", "Cubic_N", "Quartic_N")
beet = cbind(beet, N)
## Tables 4 and 5: Full polynomial analysis of variance based on raw polynomials
anova(lm(yield ~ Replicate + Linear_N + Quadratic_N + Cubic_N + Quartic_N, data = beet))
## Table 6: showing quadratic model coefficients with standard errors and confidence intervals
quadratic = lm(yield ~ Replicate + Linear_N + Quadratic_N, data = beet)
summary(quadratic)
confint(quadratic, level = 0.95)
par(mfrow = c(2, 2), oma = c(0, 0, 2, 0))
plot(quadratic, sub.caption = NA)
title(main = "Diagnostic plots for quadratic nitrogen effects model", outer = TRUE)
ggplot(beet, aes(x = nrate, y = yield)) +
ggtitle("Fig 3 Yield versus N for sugar beet with 95% confidence band") +
geom_point(shape = 1) + stat_summary(fun.y = mean, geom = "point") +
geom_smooth(method = lm, formula = y ~ poly(x, 2)) + theme_bw()
```

*agriTutorial*version 0.1.5 Index]