Fertilizer Response Across Irrigation Levels

Why an Interaction Matters

A fertilizer increase may have a different association with crop yield under different irrigation conditions. If a model includes fertilizer and irrigation only as separate additive terms, it assumes the fertilizer slope is identical at every irrigation level. An interaction allows that slope to vary by group. The practical question is not merely whether fertilizer and irrigation are each related to yield, but whether the estimated fertilizer response differs between irrigation levels.

The example throughout this tutorial is hypothetical. Imagine an experiment in which fertilizer rates and two irrigation levels are assigned to separate plots. Let \(x\) denote fertilizer rate in units of 10 kg/ha, so \(x=1\) means 10 kg/ha. Let \(I=0\) denote lower irrigation and \(I=1\) higher irrigation. Yield, \(Y\), is measured in tonnes per hectare (t/ha).

The Interaction Model

For two irrigation levels and a quantitative fertilizer predictor, a linear interaction model is

Here \(\beta_0,\beta_1,\beta_2,\beta_3\) are population regression coefficients, and \(\varepsilon_i\) is the error for plot \(i\). The interaction term is the product \(x_iI_i\). When \(I=0\), that product is zero; when \(I=1\), it equals \(x\). Thus the model gives a separate fertilizer slope for each irrigation level while retaining one equation.

The coefficients depend on the chosen reference group and units. Here, \(\beta_0\) is the mean yield at \(x=0\) under lower irrigation; \(\beta_2\) is the difference in mean yield between irrigation levels at \(x=0\). If fertilizer were measured in kg/ha rather than units of 10 kg/ha, the numerical slopes would be one tenth as large, with the same physical interpretation after units are stated.

Before using ordinary least-squares estimates and the usual \(t\)-based uncertainty calculations, the model needs appropriate conditions. The conditional mean of yield should follow the stated linear form over the fertilizer rates being analyzed. Errors from separate plots should be independent, have mean zero at each combination of predictors, and have a common variance. For exact small-sample \(t\) inference, the errors are also assumed normally distributed. In an experiment, random assignment of treatment combinations can support causal comparisons; in observational data, fitted relationships alone do not establish causation.

Worked Example: Fit the Group-Specific Lines

How the Interaction and Its Uncertainty Are Calculated

The slope formulas follow by substituting each irrigation value into the model. At \(I=0\), the conditional mean is \(\beta_0+\beta_1x\), whose coefficient on \(x\) is \(\beta_1\). At \(I=1\), it is \((\beta_0+\beta_2)+(\beta_1+\beta_3)x\), whose coefficient on \(x\) is \(\beta_1+\beta_3\). Subtracting the first slope from the second gives \(\beta_3\). This also explains why an interaction test is a test of equal fertilizer slopes.

In the hypothetical data, each cell has deviations of \(-0.2,0,0.2\) t/ha from its mean. The residual sum of squares is therefore six times the within-cell sum of squares:

There are 18 plot observations and four fitted coefficients, so the residual degrees of freedom are \(df=18-4=14\). The estimated error variance is \(MSE=SSE/df=0.48/14=0.03429\ (\text{t/ha})^2\).

For each irrigation group, the fertilizer values have mean 1 and sum of squared deviations \(3[(0-1)^2+(1-1)^2+(2-1)^2]=6\). The estimated variance of either group’s slope is \(MSE/6\), so its standard error is \(\sqrt{0.03429/6}=0.0756\) t/ha per 10 kg/ha. Because the groups are independent, the variance of the difference between their slopes is the sum of their variances. Thus the interaction standard error is

Under the stated normal-error assumptions, a 95% confidence interval for the interaction is \(\hat{\beta}_3\pm t^*SE(\hat{\beta}_3)\), where \(t^*=2.145\) for 14 degrees of freedom. The interval is \(0.50\pm2.145(0.1069)=(0.271,0.729)\) t/ha per 10 kg/ha. It describes a range of plausible values for the population difference in fertilizer slopes under this model and design.

Fitted Yields and Group Comparisons

A fitted value is the model’s estimated mean yield at specified predictor values. For instance, at \(x=1\), the predicted yields are 5.00 t/ha under lower irrigation and 6.00 t/ha under higher irrigation. The estimated difference at that fertilizer rate is 1.00 t/ha. More generally, the fitted difference between higher and lower irrigation at fertilizer value \(x\) is \(\hat{\beta}_2+\hat{\beta}_3x\). Here it is \(0.50+0.50x\): it is 0.50 t/ha at \(x=0\), 1.00 t/ha at \(x=1\), and 1.50 t/ha at \(x=2\).

A confidence interval for a fitted mean reflects uncertainty in the estimated conditional mean, not the scatter of a new individual plot around that mean. For this balanced example, the standard error of a fitted group mean at \(x=1\) is \(\sqrt{MSE/9}\), because each group has nine plots and \(x=1\) is the group’s mean fertilizer value. This is \(\sqrt{0.03429/9}=0.0617\) t/ha. Using \(t^*=2.145\), the 95% interval for the lower-irrigation fitted mean is \(5.00\pm2.145(0.0617)=(4.868,5.132)\) t/ha. The corresponding interval for the higher-irrigation fitted mean is \((5.868,6.132)\) t/ha.

Common Pitfall and Practical Interpretation

A common mistake is to read \(\hat{\beta}_1=1.00\) as the fertilizer slope for both irrigation levels. That coefficient is the slope only for the reference level \(I=0\). Under higher irrigation, the slope is \(\hat{\beta}_1+\hat{\beta}_3=1.00+0.50=1.50\). Omitting the interaction term would impose equal slopes, contrary to what this fitted model estimates.

Another error is to describe \(\hat{\beta}_3=0.50\) as a 0.50 t/ha yield increase for higher irrigation, regardless of fertilizer. It is a difference in slopes, with units of t/ha per 10 kg/ha. The fitted group difference at a particular fertilizer rate is instead \(\hat{\beta}_2+\hat{\beta}_3x\), and its units are t/ha. Always state the fertilizer rate when reporting a group contrast.

Finally, an interaction is a statement about how a modeled relationship varies across the specified groups. It does not guarantee a biological mechanism, and the linear response should not be extrapolated beyond the fertilizer rates represented by the data without additional justification. In a randomized experiment, treatment assignment can support causal interpretation for the studied conditions; otherwise, describe the result as an association.