Prof Randi Garcia
April 9, 2018
\[ {y}_{ijk}={\mu}+{\alpha}_{i}+{\beta}_{j(i)}+{\gamma}_{k}+({\alpha\gamma})_{ik}+{e}_{ijk} \]
| Source | SS | df | MS | F |
|---|---|---|---|---|
| Between | \( \sum_{i=1}^{a}tn(\bar{y}_{i..}-\bar{y}_{...})^{2} \) | \( a-1 \) | \( \frac{{SS}_{A}}{{df}_{A}} \) | \( \frac{{MS}_{A}}{{MS}_{B}} \) |
| Blocks | \( t\sum_{i=1}^{a}\sum_{j=1}^{n}(\bar{y}_{ij.}-\bar{y}_{i..})^{2} \) | \( N-a \) | \( \frac{{SS}_{B}}{{df}_{B}} \) | \( \frac{{MS}_{B}}{{MS}_{E}} \) |
| Within | \( \sum_{k=1}^{t}N(\bar{y}_{..k}-\bar{y}_{...})^{2} \) | \( t-1 \) | \( \frac{{SS}_{T}}{{df}_{T}} \) | \( \frac{{MS}_{T}}{{MS}_{E}} \) |
| Interaction | \( \sum_{i=1}^{a}\sum_{k=1}^{t}n(\bar{y}_{i.k}-\bar{y}_{i..}-\bar{y}_{..k}+\bar{y}_{...})^{2} \) | \( (a-1)(t-1) \) | \( \frac{{SS}_{AT}}{{df}_{AT}} \) | \( \frac{{MS}_{AT}}{{MS}_{E}} \) |
| Error | \( \sum_{i=1}^{a}\sum_{j=1}^{n}\sum_{k=1}^{t}({y}_{ijk}-\bar{y}_{i.k}-\bar{y}_{ij.}+\bar{y}_{i..})^{2} \) | \( (N-a)(t-1) \) | \( \frac{{SS}_{E}}{{df}_{E}} \) |
We can now imagine adding complexity to these four basic designs by including additional factors crossed with our structural factors.
Take our diabetic dogs example, and now let us add in the fact that the order of the two methods was randomly assigned. What design do we have now?
- We have an order factor and there are two levels: order 1 and order 2
- The new design is a SP/RM[2,1]
The purpose of this experiment was to study the way one species of crabgrass competed with itself and with another species for nitrogen (N), phosphorus (P), and potassium (K). Bunches of crabgrass were planted in vermiculite, in 16 Styrofoam cups; after the seeds head srouted, the plants were thinned to 20 plants per cup. Each of the 16 cups were randomly assigned to get one of 8 nutrient combinations added to its vermiculite. For example, yes-nitrogen/no-phosphorus/yes-potassium. The response is mean dry weight per plant, in milligrams.
Worms that live at the mouth of a river must deal with varying concentrations of salt. Osomoregulating worms are able to maintain reltaively constant concentration of salt in the body. An experiment wanted to test the effects of mixtures of salt water on two species of worms: Nereis virens (N) and Goldfingia gouldii (G). Eighteen worms of each species were weighted, then randomly assigned in equal numbers to one of three conditions. Six worms of each kind were placed in 100% sea water, 67% sea water, or 33% sea water. The worms were then weighted after 30, 60, and 90 minutes, then placed in 100% sea water and weighted one last time 30 minutes later. The response was body weight as percentage of initial body weight.
In an experiment, researchers wanted to compare how easy it is to remember four different kinds of words: 1) concrete, frequent: fork, brtoher, radio,… 2) concrete, infrequent: blimp, warthog, fedora, … 3) abstract, frequent: truth, anger, foolishness, … and 4) abstract, infrequent: slot, vastness, apostasy, …
Ten students in a pscyhology lab served as subject. During each of the 4 time slots, subjects heard a list of words from one of the four kinds, and then was tested for recall.
There are two possible models for chance error in models with compound within-block factors.
How can we decide?
\[ F = \frac{{MS}_{Factor}}{{MS}_{Blocks\times Factor}} \]
\[ F = \frac{{MS}_{Factor}}{{MS}_{Blocks}} \]