1. Meadowfoam Example Continuation
Suppose now we treat the variable light intensity as a quantitative
variable.
There are three possible models to look at the relationship between
seedling growth and the two predictor variables…
If we want to know whether the effect of light intensity on number
of flowers per plant depends on timing we need to include in the
model an interaction term….
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2. Interaction
Two predictor variables are said to interact if the effect that one of them has on the response depends on the value of the other.
To include interaction term in a model we simply the have to take the product of the two predictor variables and include the resulting variable in the model and an additional predictor.
Interaction terms should not routinely be added to the model. Why?
We should add interaction terms when the question of interest has to do with interaction or we suspect interaction exists (e.g., from plot of residuals versus interaction term).
If an interaction term for 2 predictor variables is in the model we should also include terms for predictor variables as well even if their coefficients are not statistically significant different from 0.
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3. Meadowfoam Example – Summary of Findings
There is no evidence that the effect of light intensity on flowers depends on timing (P-value = 0.91). That means that the interaction effect is not significant.
If interaction did exist, it is difficult to talk about the effect of light intensity or timing individually, as they would depend on the value of the other variable.
Since the interaction was not significant, consider the effect of intensity and timing from the model without the interaction term because it has smaller MSE.
For same timing, increasing light intensity decreases the mean number of flower per plant by 4.0 flowers / per plant per 100 micromol/m2/sec. 95% CI: (3.0, 5.1)
For same light intensity, beginning the light treatment early increased the mean number of flowers per plant by 12.2 flowers / plants.
95% CI (6.7, 17.6).
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4. Polynomial Regression
A “second order” model with one predictor variable is model of the form:
Y = β0 + β1X + β11X 2 + ε.
It is still a multiple linear regression model since it is linear in the
β’s. However, we can not interpret individual coefficients, since we
can’t change one variable while holding the other constant.
A “full second order model’ with 2 predictor variables is a model of
the form:
We can fit 3rd of higher order models but they get difficult to interpret.
A general rule: when fitting higher order terms, include lower order
terms in the variables in the model, even if their coefficients are not
statistically significantly different from 0.
First order terms are typically thought of as the base effect of the predictor variable on Y and higher order terms are refinements.
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5. More on Polynomial Regression
A problem that often arises with polynomial regression is that X and
X 2 are highly correlated.
To reduce this correlation and to increase numerical stability we center the data.
Centering means that we use instead of Xi1 where
The model then becomes…
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6. Rainfall Example
The data set contains cord yield (bushes per acre) and rainfall (inches) in six US corn-producing states (Iowa, Nebraska, Illinois, Indiana, Missouri and Ohio).
Straight line model is not adequate – up to 12″ rainfall yield increases and then starts to decrease.
A better model for this data is a quadratic model:
Yield = β0 + β1∙rain + β2∙rain2 + ε.
This is still a multiple linear regression model since it is linear in the β’s.
However, we can not interpret individual coefficients, since we can’t change one variable while holding the other constant…
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7. More on Rainfall Example
Examination of residuals (from quadratic model) versus year showed that perhaps there is a pattern of an increase over time.
Fit a model with year…
To assess whether yield’s relationship with rainfall depends on year we include an interaction term in the model…
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