Engineering Statistics Handbook Episode 9 Part 13 pot

The SS in a 1-way ANOVA can be split into two components, calledthe "sum of squares of treatments" and "sum of squares of error", abbreviated as SST and SSE, respectively.. The definitio

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7 Product and Process Comparisons

7.4 Comparisons based on data from more than two processes

7.4.3 Are the means equal?

7.4.3.1 1-Way ANOVA overview

Overview and

principles

This section gives an overview of the one-way ANOVA First we explain the principles involved in the 1-way ANOVA

Partition

response into

components

In an analysis of variance the variation in the response measurements is partitoned into components that correspond to different sources of variation.

The goal in this procedure is to split the total variation in the data into

a portion due to random error and portions due to changes in the values of the independent variable(s)

Variance of n

measurements

The variance of n measurements is given by

where is the mean of the n measurements

Sums of

squares and

degrees of

freedom

The numerator part is called the sum of squares of deviations from the mean, and the denominator is called the degrees of freedom.

The variance, after some algebra, can be rewritten as:

The first term in the numerator is called the "raw sum of squares" and the second term is called the "correction term for the mean" Another name for the numerator is the "corrected sum of squares", and this is usually abbreviated by Total SS or SS(Total).

7.4.3.1 1-Way ANOVA overview

http://www.itl.nist.gov/div898/handbook/prc/section4/prc431.htm (1 of 2) [5/1/2006 10:38:54 AM]

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The SS in a 1-way ANOVA can be split into two components, called

the "sum of squares of treatments" and "sum of squares of error",

abbreviated as SST and SSE, respectively

The guiding

principle

behind

ANOVA is the

decomposition

of the sums of

squares, or

Total SS

Algebraically, this is expressed by

where k is the number of treatments and the bar over the y denotes the "grand" or "overall" mean Each n i is the number of observations

for treatment i The total number of observations is N (the sum of the

n i)

Note on

subscripting

Don't be alarmed by the double subscripting The total SS can be written single or double subscripted The double subscript stems from the way the data are arranged in the data table The table is usually a

rectangular array with k columns and each column consists of n i rows

(however, the lengths of the rows, or the n i, may be unequal)

Definition of

"Treatment"

We introduced the concept of treatment The definition is: A treatment

is a specific combination of factor levels whose effect is to be compared with other treatments.

7.4.3.1 1-Way ANOVA overview

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7.4.3.2 The 1-way ANOVA model and assumptions

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ANOVA table

Total (corrected) SS N-1 The word "source" stands for source of variation Some authors prefer

to use "between" and "within" instead of "treatments" and "error", respectively

ANOVA Table Example

A numerical

example

The data below resulted from measuring the difference in resistance resulting from subjecting identical resistors to three different

temperatures for a period of 24 hours The sample size of each group was 5 In the language of Design of Experiments, we have an

experiment in which each of three treatments was replicated 5 times

Level 1 Level 2 Level 3

The resulting ANOVA table is

Example

Total (corrected) 45.349 14 Correction Factor 779.041 1 7.4.3.3 The ANOVA table and tests of hypotheses about means

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of the

ANOVA table

The test statistic is the F value of 9.59 Using an of 05, we have that

F.05; 2, 12 = 3.89 (see the F distribution table in Chapter 1) Since the test statistic is much larger than the critical value, we reject the null hypothesis of equal population means and conclude that there is a (statistically) significant difference among the population means The

p-value for 9.59 is 00325, so the test statistic is significant at that

level

Techniques

for further

analysis

The populations here are resistor readings while operating under the

three different temperatures What we do not know at this point is

whether the three means are all different or which of the three means is different from the other two, and by how much

There are several techniques we might use to further analyze the differences These are:

constructing confidence intervals around the difference of two means,

●

estimating combinations of factor levels with confidence bounds

●

multiple comparisons of combinations of factor levels tested simultaneously

●

7.4.3.3 The ANOVA table and tests of hypotheses about means

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The 829.390 SS is called the "raw" or "uncorrected " sum of squares

Step 3:

compute

SST

STEP 3 Compute SST, the treatment sum of squares.

First we compute the total (sum) for each treatment

T1 = (6.9) + (5.4) + + (4.0) = 26.7

T2 = (8.3) + (6.8) + + (6.5) = 38.6

T1 = (8.0) + (10.5) + + (9.3) = 42.8 Then

Step 4:

compute

SSE

STEP 4 Compute SSE, the error sum of squares.

Here we utilize the property that the treatment sum of squares plus the error sum of squares equals the total sum of squares

Hence, SSE = SS Total - SST = 45.349 - 27.897 = 17.45

Step 5:

Compute

MST, MSE,

and F

STEP 5 Compute MST, MSE and their ratio, F.

MST is the mean square of treatments, MSE is the mean square of error (MSE is also frequently denoted by )

MST = SST / (k-1) = 27.897 / 2 = 13.949 MSE = SSE / (N-k) = 17.452/ 12 = 1.454 where N is the total number of observations and k is the number of treatments Finally, compute F as

F = MST / MSE = 9.59

That is it These numbers are the quantities that are assembled in the

ANOVA table that was shown previously

7.4.3.4 1-Way ANOVA calculations

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discussed

later

Later on the topic of estimating more general linear combinations of means (primarily contrasts) will be discussed, including how to put

confidence bounds around contrasts 7.4.3.5 Confidence intervals for the difference of treatment means

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intervals for

the factor

level means

It can be shown that:

has a t-distribution with (N- k) degrees of freedom for the ANOVA model under consideration, where N is the total number of observations and k is the number of factor levels or groups The degrees of freedom

are the same as were used to calculate the MSE in the ANOVA table

That is: dfe (degrees of freedom for error) = N - k From this we can

calculate (1- )100% confidence limits for each i These are given by:

Example 1

Example for

a 4-level

treatment (or

4 different

treatments)

The data in the accompanying table resulted from an experiment run in

a completely randomized design in which each of four treatments was replicated five times

Total Mean

7.4.3.6 Assessing the response from any factor combination

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ANOVA

table layout

This experiment can be illustrated by the table layout for this 1-way ANOVA experiment shown below:

1. n1

2. n2

3. n3

4. n4

n t

ANOVA

table

The resulting ANOVA table is

Treatments 38.820 3 12.940 9.724

Total (Corrected) 60.112 19

Total (Raw) 979.480 20

The estimate for the mean of group 1 is 5.34, and the sample size is n1

= 5

Computing

the

confidence

interval

Since the confidence interval is two-sided, the entry /2 value for the

ttable is 5(1 95) = 025, and the associated degrees of freedom is N

-4, or 20 - 4 = 16

From the t table in Chapter 1, we obtain t.025;16 = 2.120

Next we need the standard error of the mean for group 1:

Hence, we obtain confidence limits 5.34 ± 2.120 (0.5159) and the confidence interval is

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Definition and Estimation of Contrasts

Definition of

contrasts

and

orthogonal

contrasts

Definitions

A contrast is a linear combination of 2 or more factor level means with coefficients that sum to zero.

Two contrasts are orthogonal if the sum of the products of corresponding coefficients (i.e., coefficients for the same means) adds

to zero.

Formally, the definition of a contrast is expressed below, using the notation i for the i-th treatment mean:

C = c1 1 + c2 2 + + c j j + + c k k

where

c1 + c2 + + c j + + c k = = 0

Simple contrasts include the case of the difference between two factor means, such as 1 - 2 If one wishes to compare treatments 1 and 2 with treatment 3, one way of expressing this is by: 1 + 2 - 2 3 Note that

1 - 2 has coefficients +1, -1

1 + 2 - 2 3 has coefficients +1, +1, -2

These coefficients sum to zero.

An example

of

orthogonal

contrasts

As an example of orthogonal contrasts, note the three contrasts defined

by the table below, where the rows denote coefficients for the column treatment means

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properties of

orthogonal

contrasts

The following is true:

The sum of the coefficients for each contrast is zero

1

The sum of the products of coefficients of each pair of contrasts

is also 0 (orthogonality property)

2

The first two contrasts are simply pairwise comparisons, the third one involves all the treatments

3

Estimation of

contrasts

As might be expected, contrasts are estimated by taking the same linear combination of treatment mean estimators In other words:

and

Note: These formulas hold for any linear combination of treatment

means, not just for contrasts

Confidence Interval for a Contrast

Confidence

intervals for

contrasts

An unbiased estimator for a contrast C is given by

The estimator of is 7.4.3.6 Assessing the response from any factor combination

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The estimator is normally distributed because it is a linear combination of independent normal random variables It can be shown that:

is distributed as t N-r for the one-way ANOVA model under discussion Therefore, the 1- confidence limits for C are:

Example 2 (estimating contrast)

Contrast to

estimate

We wish to estimate, in our previous example, the following contrast:

and construct a 95 percent confidence interval for C.

Computing

the point

estimate and

standard

error

The point estimate is:

Applying the formulas above we obtain

and

and the standard error is = 0.5159

Tiêu đề	One-way Anova Overview
Trường học	National Institute of Standards and Technology
Chuyên ngành	Engineering Statistics
Thể loại	Bài viết
Năm xuất bản	2006
Thành phố	Gaithersburg

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Số trang	12
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