Chapter 8: Completely Randomized DesignsScope of inferenceAn Ideal Model which allows the problems above to be solved One-way Analysis of Variance F-TestExtra Sum of Squares PrincipleNotice that there are 3 different sums of squares.Logic behind the F-testChap. 8, page 1 Chapter 8: Completely Randomized Designs A Case Study: A randomized experiment was conducted to compare lifetimes in six different diets among female mice. There were 349 mice, divided up among the treatments. We will look at this experiment in more detail as an example as we work through the next couple of chapters. Principles of good experimental design: • Randomization: female mice were randomly assigned to the six treatments. Randomization ensures no bias in the assignment of mice to treatments. It does not guarantee that the groups will be identical, but it allows us to use probability to assess whether the differences observed could have occurred by chance. • Replication – important for estimating variability within groups • Other? Scope of inference • The scope of inference is to what would have happened if all mice had been fed each diet. • The scope of inference can be expanded further if these mice can be viewed as representative of a larger population of female mice. • Since it’s an experiment, we can infer cause-and-effect if the experiment was well-run. An Ideal Model which allows the problems above to be solved fairly easily • Population distributions are normal • Population standard deviations are equal • Independent random samples from each population (a randomized experiment satisfies this assumption) This model is exactly the model for the pooled two-sample t-test when there are two groups: different means, but common standard deviation The assumption of equal standard deviations is very important and must be checked. If there are large differences in variability, this may be of interest in and of itself and the reasons for this should be addressed. Often, differing variability is caused by higher values of the variable in some groups than another. For example, the variability in lifetimes of animals is likely to be greater the longer they tend to live. Transformations (such as log) can sometimes solve this problem. Some Notation: yij: the jth sample observation from the ith population or treatment group. ni: the number of sample observations selected from population I, or assigned to the ith treatment group.Chap. 8, page 2 N: the total sample size, i.e., . in∑:iy the average of the observations in treatment group i. ..:y the average over all sample observations. The grand mean. One-way Analysis of Variance F-Test Designed to answer the question: is there evidence of a difference between any of the means? That is, we wish to test the null hypothesis 6543210:µµµµµµ=====H. The alternative hypothesis is that at least one mean is different from the others. The alternative hypothesis would include all these possibilities: • All the means are different from one another • Five means are the same and one is different • Three of the means are the same, the other three are the same but different from the first group The idea of a one-sided alternative hypothesis is meaningless with three or more groups. Your text parameterizes the full model as: ij i iyµαε=++ that is, that the jth sample observation taken from the ith population is the sum of the overall mean (shared by all populations), µ, the effect of population i on the overall variation in the observations, and an error term, ε. Here ε is normally distributed and represents the random deviation unexplained by the population differences. Under this model the null hypothesis is that all the αis are equal to zero. Testing the hypothesis of equal means relies on a general approach which we will use frequently in the rest of the course: Extra Sum of Squares Principle Full model: a general model which adequately describes the data, where the population distributions are normal with the same standard deviations, but different (possibly) means Reduced model: a special case of the full model obtained by imposing the restriction of the null hypothesis, where the population distributions are normal with the same standard deviations, and the same means. The general idea is that we “fit” both these models to the data. Each model gives a predicted value for every case. The full model uses each observation’s group mean as the predicted value. The reduced model uses the mean of all the observations together. We then measure how well the data fit the models by computing the sum of squared residuals. The full model can fit no worse than the reduced model because the reduced model is a special case of the full model.Chap. 8, page 3 So, the predicted responses are Group 1 2 3 4 5 6 Full 1y 2y 3y 4y 5y 6y Reduced ..y ..y ..y ..y ..y ..y Example: To illustrate these calculations, we’ll use a small hypothetical example, with 3 groups and 10 observations in all. Group 1: 10.7 13.2 15.7 = 3 1n1y = 13.2 = 2.500 1sGroup 2: 12.1 14.2 16.0 16.5 = 4 2n2y = 14.7 = 1.995 2sGroup 3: 20.9 24.4 27.3 = 3 3n3y = 24.2 = 3.205 3sTotal: n = 10 ..y = 17.1 = 2.535 ps ..y is called the “grand mean” and is the mean of all 10 observations. Group ObsResponse Predicted (reduced model) Residual (reduced model) ijyy=− Squared residual (reduced) Predicted (full model) Residual (full model) .ij iyy=− Squared residual (full) 1 1 10.7 17.1 -6.4 40.96 13.2 -2.5 6.25 1 2 13.2 17.1 -3.9 15.21 13.2 0.0 0.00 1 3 15.7 17.1 -1.4 1.96 13.2 2.5 6.25 2 1 12.1 17.1 -5.0 25.00 14.7 -2.6 6.76 2 2 14.2 17.1 -2.9 8.41 14.7 -0.5 0.25 2 3 16.0 17.1 -1.1 1.21 14.7 1.3 1.69 2 4 16.5 17.1 -0.6 0.36 14.7 1.8 3.24 3 1 20.9 17.1 3.8 14.44 24.2 -3.3 10.89 3 2 24.4 17.1 7.3 53.29 24.2 0.2 0.04 3 3 27.3 17.1 10.2 104.04 24.2 3.1 9.61 Total 264.88 44.98 Extra sum of squares = Residual sum of squares (reduced) – Residual sum of squares (full) = 264.88 - 44.98 = 219.9 The residual sum of squares for a model represents the variability in the original data which is not explained by the model. The extra sum of squares therefore represents the amount of the unexplained variability in the reduced model that is explained by the full
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