anova

Dr. Ian Vallance

Dr. Ricky TomanekImpact Laboratories

Ian Galloway FRSS,

Lean/six sigma green belt

Introduction to Introduction to ANANalysis alysis OOf f VAVAriance (ANOVA)riance (ANOVA)

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Course content1. What is ANOVA

2. Different types of ANOVA

3. ANOVA Theory

4. Worked example in excela) Generating the data b) Explanation/Interpretation of outputc) Rules for accepting/rejecting null hypothesis

5. Supplemental testing

6. Summary – ANOVA in Excel

7. Summary

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1 What is ANOVA

ANOVA is a general technique that can be used to test the hypothesis that the means among two or more groups are equal, under the assumption that the sampled populations are normally distributed.

Analysis of variance can be used to test differences among several means for significance without increasing the Type I error rate.

The t-test is designed to test the hypothesis that 2 means could be from the same population of data

But what if we want to compare more than 2 means at the same time?

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2 Different types of ANOVATo begin, let us consider the effect of temperature on a passive component such as a resistor.

We select three different temperatures and observe their effect on the resistors.

This experiment can be conducted by measuring all the participating resistors before placing n resistors each in three different ovens.

Each oven is heated to a selected temperature. Then we measure the resistors again after, say, 24 hours and analyze the responses, which are the differences between before and after being subjected to the temperatures.

The temperature is called a factor.

The different temperature settings are called levels. In this example there are three levels or settings of the factor Temperature.

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Different types of ANOVAWhat is a factor?

A factor is an independent treatment variable whose settings (values) are controlled and varied by the experimenter.

The intensity setting of a factor is the level. Levels may be quantitative numbers or, in many cases, simply "present" or "not present" ("0" or "1").

In the experiment, there is only one factor, temperature, and the analysis of variance that we will be using to analyze the effect of temperature is called a one-way or one-factor ANOVA.

The 1-way ANOVA

The 2-way or 3-way ANOVA

We could have opted to also study the effect of positions in the oven. In this case there would be two factors, temperature and oven position. Here we speak of a two-way or two-factor ANOVA.

Furthermore, we may be interested in a third factor, the effect of time. Now we deal with a three-way or three-factorANOVA.

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3 ANOVA Theory

The theory of ANOVA is long, complicated and detailed and will NOT be looked at in this course.If you do want to learn more try the following web pages:

http://davidmlane.com/hyperstat/intro_ANOVA.html

http://itl.nist.gov/div898/handbook/prc/section4/prc43.htm

http://www.experiment-resources.com/anova-test.html

http://www.chem.agilent.com/cag/bsp/products/gsgx/Downloads/pdf/one_way_anova.pdf

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4 (a) A worked example in excel 2007

In this example of a one way ANOVA we will calculate all the components of the

ANOVA without explaining the theory behind the formulas used.

The main objectives of this exercise are to learn about the typical layout of

ANOVA output (the format will look very similar in excel)

and to learn how to interpret the output.

And finally how to carry out ANOVA using excel

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A worked example in excel 2007

Aspirin Tylenol Placebo

3 2 2

5 2 1

3 4 3

5 4 2

How many factors?

How many levels?

How many subjects in total?

In a hypothetical experiment, aspirin, Tylenol, and a placebo were tested to see how much pain relief each provides. Pain relief was rated on a five-point scale. Four subjects were tested in each group and their data are shown below:

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(1) Data for analysis

(2) Use the data

tab

(3) Data analysis toolpak

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Select Data Analysis

Why are we selecting this

option?

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Input range is the data to be

analysed

If the data columns have labels tick this

box

Alpha is the confidence Level i.e.

0.05 = 95%CL

Output options – where do you want to see the results

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4 (b)A worked example Interpretation of output

Now we have an output – We can deconstruct each result and explain the final result

Anova: Single Factor

SUMMARY

Groups Count SumAverag

eVarianc

eAspirin 4 16 4 1.33Tylenol 4 12 3 1.33Placebo 4 8 2 0.67

ANOVASource of Variation SS df MS F P-value F crit

Between Groups 8 2 4 3.6 0.07 4.26Within Groups 10 9 1.11

Total 18 11

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A worked example Interpretation of output



Total 18 11

Total Sum of SquaresThe variation among all the subjects in an experiment is measured by what is called total sum of squares or SST.

SST is the sum of the squared differences of each score from the mean of all the scores.

Letting GM (standing for "grand mean") represent the mean of all scores, then

SST = Σ(X - GM)²

where GM = ΣX/N and N is the total number of subjects in the experiment

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For the example data:

N = 12

GM = (3+5+3+5+2+2+4+4+2+1+3+2)/12 = ?

SST = (3-3)²+(5-3)²+(3-3)²+(5-3)² + (2-3)²+(2-3)²+(4-3)²+(4-3)² + (2-3)²+(1-3)²+(3-3)²+(2-3)² = ?

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Total 18 11

Sum of Squares Between Groups

The sum of squares due to differences between groups (SSB) is computed according to the following formula:

where ni is the sample size of the ith group and Mi is the mean of the ith group, and GM is the mean of all scores in all groups.

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If the sample sizes are equal then the formula can be simplified somewhat:

SSB = nΣ(Mi - GM)²

For the example data,

M1 = (3+5+3+5)/4 = 4

M2 = (2+4+2+4)/4 = 3

M3 = (2+1+3+2)/4 = 2

GM = 3

n = 4

SSB = 4[(4-3)² + (3-3)² + (2-3)²] = ?

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Total 18 11

Sum of Squares Error (Sum of Squares within groups)The sum of squares error is the sum of the squared differences between the individual scores and their group means.

SSE = SSE1 + SSE2 + ... + SSEa

SSE1 = Σ(X - M1)² ; SSE2 = Σ(X - M2)² SSEa = Σ(X - Ma)²

where M1 is the mean of Group 1, M2 is the mean of Group 2, and Ma is the mean of Group a.

Or you could Simply say! SSwithin = Sstotal - SSbetween

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Total 18 11

dfbetween = a - 1 = 3 - 1 = 2

dfwithin = N - a = 12 - 3 = 9

dftotal = N - 1 = 12 - 1 = 11

a is the number groups N is the total number of subjects. dftotal = dfgroups + dferror

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Total 18 11

Mean squares are estimates of variance and are computed by dividing the sum of squares by the degrees of freedom.

The mean square for groups (4.00) was computed by dividing the sum of squares for groups (8.00) by the degrees of freedom for groups (2).

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Total 18 11

The F ratio is computed by dividing the mean square for between groups by the mean square for within groups.

In this example, F = 4.000/1.111 = 3.60

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Total 18 11

The probability value It is the probability of obtaining an F as large or larger than the one computed in the data assuming that the null hypothesis is true.

It can be computed from an F table.

The df for groups (2) is used as the degrees of freedom in the numerator and the df for error (9) is used as the degrees of freedom in the denominator.

The probability of an F with 2 and 9 df as larger or larger than 3.60 is 0.071

Fcrit is the highest value of F that can be obtained without rejecting the null hypothesis (obtained from F-test tables for 2&9 DF)

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4(c) A worked example Interpretation of output

Interpreting the Anova One Way test results

If Then

test statistic > critical valueReject the null hypothesis(i.e. F> Fcrit)

test statistic < critical value

Accept the null hypothesis(i.e. F< Fcrit)

p value < Reject the null hypothesis

p value > Accept the null hypothesis

Using the above table and the results from the example is there an indication of a significant difference in the means?

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5 Introduction to Tests Supplementing a One-factor Between-Subjects ANOVA

Unfortunately, when the analysis of variance is significant and the null hypothesis is rejected.

The only valid inference that can be made is that at least one population mean is different from at least one other population mean.The analysis of variance does not reveal which population means differ from which others

Consequently, further analyses are usually conducted after a significant analysis of variance.

These further analyses almost always involve conducting a series of significance tests.

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Additional info - PCER (not in training)

The probability that a single significance test will result in a Type I error is called the pre-comparison error rate (PCER).

The probability that at least one of the tests will result in a Type I error is called the experiment wise error rate(EER).

Statisticians differ in their views of how strictly the EER must be controlled. Some statistical procedures provide strict control over the EER whereas others control it to a lesser extent.

Naturally there is a trade off between the Type I and Type II error rates.

The more strictly the EER is controlled, the lower the power of the significance tests

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Additional info - PCER (not in training)

When a series of significance test is conducted, the experimentwise error rate (EER) is the probability that one or more of the significance tests results in a Type 1 error. If the comparisons areindependant, then the experimentwise error rate is:

where

αew is experimentwise error rate

αpc is the per-comparison error rate

and c is the number of comparisons. For example, if 5 independent comparisons were each to be done at the .05 level then the probability that at least one of them would result in a Type I error is:

1 - (1 - .05)5 = 0.226.

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Introduction to Tests Supplementing a One-factor Between-Subjects ANOVA

The "Honestly Significantly Different" (HSD) test proposed by the statistician John Tukey is based on what is called the “studentised range distribution."

To test all pairwise comparisons among means using the Tukey HSD, compute t for each pair of means using the formula:

where Mi - Mj is the difference between the ith and jth means, MSE is the Mean Square within, and nh is the harmonic mean of the sample sizes of groups i and j.

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Introduction to Tests Supplementing a One-factor Between-Subjects ANOVA

The critical value of ts is determined from the distribution of the studentised range.

The number of means in the experiment is used in the determination of the critical value, and this critical value is used for all comparisons among means.

Typically, the largest mean is compared with the smallest mean first.

If that difference is not significant, no other comparisons will be significant either, so the computations for these comparisons can be skipped.

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Summary ANOVA in Excel

Anova: Single Factor

This tool performs a simple analysis of variance on data for two or more samples.

The analysis provides a test of the hypothesis that each sample is drawn from the same underlying probability distribution against the alternative hypothesis that underlying probability distributions are not the same for all samples.

If there are only two samples, you can use the worksheet function TTEST.

With more than two samples, there is no convenient generalization of TTEST, and the Single Factor Anova model can be called upon instead.

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Anova: Two-Factor with Replication

This analysis tool is useful when data can be classified along two different dimensions.

For example, in an experiment to measure the height of plants, the plants may be given different brands of fertilizer (for example, A, B, C) and might also be kept at different temperatures (for example, low, high).

For each of the six possible pairs of {fertilizer, temperature}, we have an equal number of observations of plant height.

Using this Anova tool, we can test:

Whether the heights of plants for the different fertilizer brands are drawn from the same underlying population. Temperatures are ignored for this analysis.

Whether the heights of plants for the different temperature levels are drawn from the same underlying population. Fertilizer brands are ignored for this analysis.

Summary ANOVA in excel

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Whether, having accounted for the effects of differences between fertilizer brands found in the first bulleted point ,and differences in temperatures found in the second bulleted point, the six samples representing all pairs of {fertilizer, temperature} values are drawn from the same population.

The alternative hypothesis is that there are effects due to specific {fertilizer, temperature} pairs over and above the differences that are based on fertilizer alone or on temperature alone.

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Anova: Two-Factor Without Replication

This analysis tool is useful when data is classified on two different dimensions as in the Two-Factor case With Replication.

However, for this tool it is assumed that there is only a single observation for each pair (for example, each {fertilizer, temperature} pair in the preceding example).

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Summary• To compare 2 or more means in a single test we use

ANOVA

• The type of ANOVA test to use is decided by the number of FACTORS in the experiment

• The ANOVA will only tell whether there is a significant difference and gives no information on which mean(s) are different

• Further pairwise comparisons of the means are required to gain further information on which mean(s) are different

• Pairwise testing of means can increase the probability of type 1 errors

anova

Documents

anova anovais

factor temperature

aoneway oronefactor

example data

final result anova

excel summary

effect of temperature

data tab