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…. week 11

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. week 11

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). week 11

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. week 11

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… week 11

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… week 11

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… week 11