1. An industrial psychologist was interested in decreasing the time required to assemble an electronic component. Three types of assembly fixtures, including the one currently in use (control), were evaluated. For this study, six factories were selected randomly from a large population. The six factories were assigned randomly to the three assembly fixtures, with the constraint that each type of fixture was tested in two factories. Five workers in each factory were randomly selected from the entire workforce to participate in the study. After a three-week familiarization period, the experimenters measured the assembly rate (units assembled per hour) for each participant. In this experiment, fixture is fixed and factory is random, and factory is nested within fixture. The data are stored in the data frame electronics which can be loaded and inspected with the following commands:

load(url("http://pnb.mcmaster.ca/bennett/psy710/datasets/electronics.rda") )
sapply(electronics,class)
xtabs(~fixture + factory,data=electronics) # factory nested in fixture

1. Evaluate the statistical significance of the fixed effect of fixture and the random effect of factory.

2. Compute association strength for fixture and factory.

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2. An experiment analyzed the glycogen content of rat livers. There were three experimental treatment groups: a control group, a group treated with compound 217, and a group treated with compound 217 plus sugar. Six rats were selected randomly from a large population, and two were assigned randomly to each treatment. The experimenters took three samples of liver (i.e., three pieces of tissue) from each rat, and then two measures of glycogen were made on each sample for a total of 36 measures. The data are stored in the data frame liver which can be loaded with the following command:

load(url("http://pnb.mcmaster.ca/bennett/psy710/datasets/liver.rda") )
sapply(liver,class)
##  glycogen treatment       rat    sample   measure 
## "integer"  "factor"  "factor"  "factor"  "factor"
# summary(liver)
# xtabs(~treatment + rat,data=liver) # rat nested in treatment
# xtabs(~rat+sample,data=liver) # sample nested in rat

In this experiment, glycogen is the dependent variable, treatment is a fixed factor and rat and sample are random factors. Note that rat is nested within treatment and sample is nested within rat. The factor sample contains 18 unique levels, with each level representing one of the three samples taken from the six rats. The data frame also contains the factor measure, which you can think of as the ID for each glycogen measure (similar to the way subjects often are identified by an ID).

1. Evaluate the effect of treatment on glycogen while ignoring the random effects and nested structure of the data.

2. Evaluate the statistical significance of the fixed effect of treatment on glycogen using a model that includes the random/nested factors. Does the effect of treatment differ from the one estimated in the previous question? Why or why not?

3. Evaluate the statistical significance of the random effects.

4. Estimate the variance components.