Readings 1.1, 1.2, 1.4, Rmarkdown
- [Golf] We can imagine someone’s golf score as being a function of many factors,
golf score = f(driver type, type of ball, ...)In theory, we could manipulate these factors to see how they influenced golf score. If we considered only two factors at a time, each with two possible levels, this would be called a \(2^2\) design.
- We can visualize the 4 possible configurations as corners of a square The golf score is the height of the plane.
Mathematically,
\[ y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \epsilon \]
Interactions
- It’s possible that the effect of one factor depends on the value of the other – this called an interaction between the two factors. If this happens, then the slopes along the edges are no longer parallel. The previous formula cannot capture this. Instead, we need,
\[ y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \beta_{12}x_{1}x_{2} + \epsilon \]
because now the slopes can change depending on the value of the other factor.
- For example, rearranging terms, we can see that the slope for \(x_1\) depends on the value of \(x_2\),
\[ y = \beta_{0} + \left(\beta_{1} + \beta_{12}x_2\right)x_{1} + \beta_{2}x_{2} + \epsilon. \]
Can you write an expression showing how the slope for \(x_{2}\) depends on \(x_{1}\)?
- For each configuration of factors, it is better to play several rounds of golf. The more rounds we play, the better our estimates of the effects for each factor. This is a special case of what we discussed in the last notes; the more replicates, the better our estimates.
More than 2 factors
Suppose we want to see how K different binary factors influence golf score. We can no longer visualize the effects as corners of a square, but we can still collect samples for each configuration of factors. This is called a \(2^K\) experiment.
A challenge is that for large K, this means collecting lots of samples
- K = 3 means 8 configurations
- K = 4 means 16
- etc.
- Experimental design is often used in characterizing a process; i.e., how do each of the knobs affect the outcome? Alternatively, we may ask a simpler question – are there knobs that have no effect on the outcome? This is called factor screening. An example is the soldering experiment.
Sometimes we care more about optimization. In this case, we don’t care so much about how each factor influences an outcome; we just want a combination of factors that maximizes it. We can visualize the outcome of hte process as a function of several continuous variables.
Intuitively, our experimentation should proceed by first making a preliminary test and then proceeding in the direction of the max. This intuition is formalized in response surface methodology.
