1. 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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2. 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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3. 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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4. Indicator Variables
Often, a data set will contain categorical variables which are
potential predictor variables.
To include these categorical variables in the model we define
dummy variables.
A dummy variable takes only two values, 0 and 1.
In categorical variable with j categories we need j-1 indictor variables.
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5. Meadowfoam Example
Meadowfoam is a small plant found in the US Pacific Northwest.
Its seed oil is unique among vegetable oils for its long carbon
strings, and it is nongreasy and highly stable. A study was conducted
to find out how to elevate meadowfoam production to a profitable
crop. In a growth chamber, plants were grown under 6 light
intensities (in micromol/m^2/sec) and two timings of the onset of
the light treatment, either late (coded 0) or early (coded 1). The
response variable is the average number of flowers per plant for 10
seedlings grown under each of the 12 treatment conditions.
This is an example of an experiment in which we can make causal conclusions.
There are two explanatory variables, light intensity and timing.
There are 24 data points, 2 at each treatment combination.
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6. Question of Interests
What is the effect of timing on the seedling growth?
What are the effects of the different light intensity?
Does the effect of intensity depend on timing?
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7. Indicator Variables in Meadowfoam Example
To include the variable time in the model we define a dummy
variable that takes the value 1 if early timing and the value 0 if late
timing.
The variable intensity has 6 levels (150, 300, 450, 600, 750, 900). We will treat these levels as 6 categories.
It is useful to do so if we expect a complex relationship between
response variable and intensity and if the goal is to determine which
intensity level is “best”.
The cost in using dummy variables is degrees of freedom since we need multiple dummy variables for each of the multiple categories.
We define the dummy variables as follows….
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8. Partial F-test
Partial F-test is designed to test whether a subset of β’s are 0 simultaneously.
The approach has two steps. First we fit a model with all predictor variables. We call this model the “full model”.
Then we fit a model without the predictor variables whose coefficients we are interested in testing. We call this model the “reduced model”.
We then compare the SSReg and RSS in these two models….
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9. Test Statistic for Partial F-test
To test whether some of the coefficients of the explanatory variables
are all 0 we use the following test statistic: .
Where Extra SS = RSSred - RSSfull, and
Extra df = number of parameters being tested.
To get the Extr SS in SAS we can simply fit two regressions
(reduced and full) or we can look at Type I SS which are also called
Sequential Sum of Squares.
The Sequential SS gives the additional contribution to SSR each
variable gives over and above variables previously listed.
The Sequential SS depends on which order variables are stated in
model statement; the variables whose coefficients we want to test
must be listed last.
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10. 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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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 on Y, as it varies with timing.
Since the interaction was not significant, we remove it from the model.
For same timing, increasing light intensity by 100 micromol/m2/sec decreases the mean number of flower per plant by 4.0 flowers / per plant. 95% CI: (-5.1, -3)
For same light intensity, beginning the light treatment early increases the mean number of flowers per plant by 12.2 flowers / plants.
95% CI (6.7, 17.6).
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