Example
A study was conducted to assess the effects of feedback in a repetitive industrial task. The task was to grind a metal piece to a specified size and shape. Eighteen male workers were divided randomly into three groups. All subjects were given the same introduction to the task.
After beginning the experimental period, the subjects in one group received no feedback about the task, those in the second group were given vague and intermittent feedback, and subjects in the third group were given accurate and continuous feedback. The response consisted of a measure of the value, in dollars, added to production by each subject during the experimental period. This measure was a function of the number of pieces produced, the accuracy of the grinding operation and the amount of reworking necessary in the remaining stages of production. One worker became ill during the study, and his data were dropped. The data, found in SASDATA.FEEDBACK, are:
Type of Feedback | ||
None | Vague | Accurate |
40.85 | 38.32 | 48.59 |
35.21 | 40.26 | 40.71 |
38.17 | 47.47 | 45.33 |
43.96 | 44.10 | 43.76 |
34.88 | 40.09 | 46.41 |
42.67 | 44.19 |
For the present data, i=3, n_{1}=5, and n_{2}=n_{3}=6; is the population mean for the no feedback group, and and are the population means for the vague and accurate feedback groups, respectively.
The least squares estimator of is just the mean of the observations from population i:
The corresponding estimator of is the pooled variance estimator where S^{2}_{i} is the sample variance of the observations from population i.Let's check out the fit of the model to the feedback data.
The residuals
are used to check the fit.
Here's how to check the fit of the model to the feedback data.
The question researchers most often ask concerning the means model is "Are the population means all equal?"
Formally, the hypotheses are
H_{0}: | = | = | = | |||||
H_{a}: | Not all the population means | |||||||
are equal. |
These hypotheses are tested using the F statistic. To see what the F statistic is all about, we first need to learn about partitioning the variation in the response into different components, in what is known as the Analysis of Variance (aka ANOVA).
The total variation in the responses is measured by the total sum of squares, SSTO:
This variation can be broken into two components: the variation explained by the model, the model sum of squares and the variation left unexplained by the model, the error sum of squares Further, these components add: SSTO=SSM+SSE.Each SS has associated with it a number, called its degrees of freedom, which counts the number of independent pieces of data going into the SS. The df for SSTO, SSM and SSE are n-1, k-1 and n-k. These df add exactly like their SS.
The mean square is the SS divided by its df. Thus MSM=SSM/(k-1), and MSE=SSE/(n-k).
The test statistic for testing equality of population means is the F statistic F=MSM/MSE. It is compared with its distribution under H_{0}, which is an F_{k-1,n-k} distribution. Large values of F support H_{a} over H_{0}.
The information about SS, df, MS and the F test is summarized in an ANOVA table. Let's have a look...
Pairs of population means can be compared using t tests or confidence intervals.
Consider the confidence interval we've just shown for comparing two population means. This type of comparison is called a pairwise comparison, since when we set the confidence level we are only concerned with that particular comparison. If we are doing a lot of these comparisons, we can run into problems interpreting the confidence levels. These problems have to do with
Two multiple comparison procedures which control the overall error rate for all comparisons made are the Bonferroni and Tukey procedures.
When studentized by dividing by estimated standard error, this distribution is called the studentized range distribution. It is suitable for use in data snooping. A level L Tukey interval for is
where q_{L,k,n-k} is the L^{th} quantile of the studentized range distribution.The one-way effects model is the one-way means model parametrized to emphasize the deviations of population means from an overall mean. It is written
where is an overall mean for all populations, and is the effect due to the i^{th} population.Example
Four types of highway surface are being tested for durability. Engineers obtained 10 different sites on existing highways to test these surfaces. Since the sites are on different types of highways, the engineers decided to divide each site into four equal sections and randomly assign one surface to each section in such a way that all four surface types appear at each site. In reality, the test sites were monitored periodically and a number of measures of wear were taken on each occasion.
The response we will consider is an index of severity of wear, coded on a scale of 0 (no wear) to 100 (severe wear). The data are found in SASDATA.ASPHALT.
Let's look at the data.
One useful model for data of this type is the randomized complete block model:
Notice that this is an effects model with two factors: blocks, represented by the effects and treatments, represented by the effects. Notice also the model is additive: the effects add.
The least squares estimators of the parameters are:
The fitted values are
The residuals are, as usual, the observed minus the fitted values,
The fit is checked by
Let's check out the fit for the asphalt data ourselves.
We test
= | = | = | = | ||||||
Not all the population effects | |||||||||
are | 0. |
As for the one-way model, the ANOVA table shows sums of squares, degrees of freedom and mean squares for the RCB model.
Let's look at the analysis for the asphalt data.
H_{0}: | = | ||
H_{a}: |
As for the one-way model, we may use either the Bonferroni or the Tukey procedure to compare more than one pair of means.
Consider the ANOVA table for the asphalt data considered as a RCBD:
and as a one-way model:
Now consider what this does for estimation. Here is a level 0.95 confidence interval for the difference in mean wear between asphalt types 2 and 3 computed from the one-way model:
(4.915,20.085)
And here is a level 0.95 confidence interval for the difference in mean wear between asphalt types 2 and 3 computed from the RCB model:(8.963,16.037).
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