Analysis of Variance

Analysis of Variance

CSLU 2850.Lo1Spring 2008

Cameron [email protected]

Fordham University

May contain work from the Creative Commons.


• Analysis of Variance In the Statistical Inference section, we studied

how a two-sample t-test can be used to determine whether a significant difference exists between the samples.

Now, we will use Analysis of Variance to compare the means of two or more samples.


• Null and Alternative Hypotheses H0: μ1 = μ2 = … = μn

HA: Not all of the μi are equal.

Here, there are n samples and μi is the mean of the ith sample.


• Means Model The model we use for analysis of variance is

called the Means Model

Here, μi is the mean of the ith sample and ε is the random error following a normal distribution with mean 0 and variance σ2.

iy


• Necessary Assumptions The errors are normally distributed. The errors are independent. The errors have constant variance, σ2.


• Normal Distribution of the Residuals To test this assumption, create a normal plot of

the residuals. If the residuals follow a normal distribution, they

should fall evenly along the normal probability plot.


• Independent errors This final assumption should only be considered

when there is a defined order of the observations.

We would like to make sure that the residual of a data point is not influenced by the surrounding data points.


• Constant variance assumption Next, we will look at the assumption of constant

variance in the residuals. To view this graphically, plot the residual values against the predicted values.

If there is an increase in variance, at any point along the line, then there is reason to doubt the assumption of constant variance.

Outliers may have a significant impact on this test.


• AoV and Regression Analysis of Variance is a special form of

Regression. In the case of Analysis of Variance, the predictor

variables are discrete, rather than continuous. To express an Analysis of Variance in terms of

Regression, we must reformulate the model.


• Effects Model We will make modifications to the Means

Model.

Where, μ is the mean term, αi is the effect term for the ith sample, and ε is a normally distributed error term with mean 0 and variance σ2.

iy


• Two-way Analysis of Variance One-way Analysis of Variance compares several

samples corresponding to a factor. Two-way Analysis of Variance uses two factors,

e.g. medication and dose in a medical experiment.


• Means Model

Where y is the response variable, uij is the mean for the ith level of one factor and jth level of the second factor, and eijk is the error for the ith level for the first factor and jth level for the second factor and kth replicate.

ijkijijky


• Means Model More commonly, the Means Model is represented as:

Where y is the response variable, αi is the effect of the ith level of the first factor, βj is the effect of the jth level of the second factor , αβij is the interaction between the two factors, and eijk is the error for the ith level for the first factor and jth level for the second factor and kth replicate.

ijkijjiijky

Homework(Always Due in One WeekAlways Due in One Week)

• Read Chapter 10.• Complete Chapter 10, pg. 400:1 (a-d), 4


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Analysis of Variance