A model for data is a specification of the statistical distribution for the data. For example, the number of heads in ten tosses of a fair coin would have a Binomial(10,.5) distribution, where .5 gives the probability of a success and 10 is the number of trials. In this instance, the distribution depends on two numbers, called parameters: the success probability and the number of trials. For ten tosses of a fair coin, we know both parameters. In the analysis of
3.3 Models and Parameters 35 5 4 3 2 1 2.0 1.5 1.0 Treatments Log10 lifetime
Figure 3.1:Box-plots oflog10times till failure of a resin under five different temperature stresses, using Minitab.
1.5 2 2.5 3 120 140 160 180 200 220 240 Temperature M e a n l o g l i f e t i m e
Figure 3.2:Averagelog10time till failure versus temperature, with linear regression line added, using MacAnova.
experimental data, we may posit several different models for the data, all with unknown parameters. The objectives of the experiment can often be described as deciding which model is the best description of the data, and making inferences about the parameters in the models.
36 Completely Randomized Designs
Our models for experimental data have two basic parts. The first part describes the average or expected values for the data. This is sometimes called a “model for the means” or “structure for the means.” For example,
Model for the
means consider the birch tree weights from Example 3.1. We might assume that all the treatments have the same mean response, or that each treatment has its own mean, or that the means in the treatments are a straight line function of the treatment pH. Each one of these models for the means has its own parameters, namely the common mean, the five separate treatment means, and the slope and intercept of the linear relationship, respectively.
The second basic part of our data models is a description of how the data vary around the treatment means. This is the “model for the errors”
Model for the
errors or “structure for the errors”. We assume that deviations from the treatment
means are independent for different data values, have mean zero, and all the deviations have the same variance, denoted byσ2.
σ2
This description of the model for the errors is incomplete, because we have not described the distribution of the errors. We can actually go a fair way with descriptive statistics using our mean and error models without ever
Normal distribution of errors needed for inference
assuming a distribution for the deviations, but we will need to assume a dis- tribution for the deviations in order to do tests, confidence intervals, and other forms of inference. We assume, in addition to independence, zero mean, and constant variance, that the deviations follow a Normal distribution.
The standard analysis for completely randomized designs is concerned with the structure of the means. We are trying to learn whether the means
Standard analysis
explores means are all the same, or if some differ from the others, and the nature of any
differences that might be present. The error structure is assumed to be known, except for the variance σ2, which must be estimated and dealt with but is
otherwise of lesser interest.
Let me emphasize that these models in the standard analysis may not be the only models of interest; for example, we may have data that do not
Standard analysis is not always appropriate
follow a normal distribution, or we may be interested in variance differences rather than mean differences (see Example 3.4). However, the usual analysis looking at means is a reasonable place to start.
Example 3.4 Luria, Delbr ¨uck, and variances
In the 1940s it was known that some strains of bacteria were sensitive to a particular virus and would be killed if exposed. Nonetheless, some members of those strains did not die when exposed to the virus and happily proceeded to reproduce. What caused this phenomenon? Was it spontaneous mutation, or was it an adaptation that occurred after exposure to the virus? These two competing theories for the phenomenon led to the same average numbers
3.3 Models and Parameters 37
of resistant bacteria, but to different variances in the numbers of resistant bacteria—with the mutation theory leading to a much higher variance. Ex- periments showed that the variances were high, as predicted by the mutation theory. This was an experiment where all the important information was in the variance, not in the mean. It was also the beginning of a research collab- oration that eventually led to the 1969 Nobel Prize for Luria and Delbr¨uck.
There are many models for the means; we start with two basic models. We have g treatments and N units. Let yij be the jth response in the ith
treatment group. Thusi runs between 1 and g, and j runs between 1 and ni,
in treatment groupi. The model of separate group means (the full model) as- Separate means model
sumes that every treatment has its own mean responseµi. Combined with the
error structure, the separate means model implies that all the data are inde- pendent and normally distributed with constant variance, but each treatment group may have its own mean:
yij ∼ N(µi, σ2) .
Alternatively, we may write this model as
yij = µi+ ǫij ,
where theǫij’s are “errors” or “deviations” that are independent, normally
distributed with mean zero and varianceσ2.
The second basic model for the means is the single mean model (the
reduced model). The single mean model assumes that all the treatments have Single mean model
the same meanµ. Combined with the error structure, the single mean model
implies that the data are independent and normally distributed with meanµ
and constant variance,
yij ∼ N(µ, σ2) .
Alternatively, we may write this model as
yij = µ + ǫij ,
where theǫij’s are independent, normally distributed errors with mean zero
and varianceσ2.
Note that the single mean model is a special case or restriction of the Compare reduced model to full model
group means model, namely the case when all of the µi’s equal each other.
Model comparison is easiest when one of the models is a restricted or reduced form of the other.
38 Completely Randomized Designs
We sometimes express the group meansµiasµi = µ⋆+ αi. The constant
µ⋆ is called the overall mean, andαiis called theith treatment effect. In this
Overall meanµ⋆
and treatment effectsαi
formulation, the single mean model is the situation where all theαi values
are equal to each other: for example, all zero. This introduction of µ⋆ and
αiseems like a needless complication, and at this stage of the game it really
is. However, the treatment effect formulation will be extremely useful later when we look at factorial treatment structures.
Note that there is something a bit fishy here. There are g means µi,
one for each of the g treatments, but we are using g + 1 parameters (µ⋆
and theαi’s) to describe theg means. This implies that µ⋆ and theαi’s are
Too many
parameters not uniquely determined. For example, if we add 15 to µ⋆ and subtract 15
from all theαi’s, we get the same treatment meansµi: the 15’s just cancel.
However, αi − αj will always equal µi − µj, so the differences between
treatment effects will be the same no matter how we defineµ⋆.
We got into this embarrassment by imposing an additional mathematical structure (the overall meanµ⋆) on the set ofg group means. We can get out of
this embarrassment by deciding what we mean byµ⋆; once we knowµ⋆, then we can determine the treatment effects αi by αi = µi− µ⋆. Alternatively,
Restrictions make treatment effects well defined
we can decide what we mean byαi; then we can getµ⋆ byµ⋆ = µi − αi.
These decisions typically take the form of some mathematical restriction on the values for µ⋆orα
i. Restricting µ⋆ orαi is really two sides of the same
coin.
Mathematically, all choices for definingµ⋆ are equally good. In prac- tice, some choices are more convenient than others. Different statistical soft- ware packages use different choices, and different computational formulae
Differences of treatment effects do not depend on restrictions
use different choices; our major worry is keeping track of which particular choice is in use at any given time. Fortunately, the important things don’t
depend on which set of restrictions we use. Important things are treatment
means, differences of treatment means (or equivalently, differences ofαi’s),
and comparisons of models.
One classical choice is to defineµ⋆as the mean of the treatment means:
µ⋆ =
g
X
i=1
µi/g .
For this choice, the sum of the treatment effects is zero:
Sum of treatment effects is zero g X i=1 αi= 0 .