anova test in mathematics, Schemes and Mind Maps of Mathematical Statistics

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Analysis of Variance - ANOVA

Eleisa HeronEleisa Heron

Introduction •

Analysis of variance (ANOVA) is a method for testing the hypothesis that there is nodifference between two or more population means (usually at least three)

ANOVA

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Often used for testing the hypothesis that there is no difference between a number oftreatments

Why Not Use t-test Repeatedly? •

The t-test, which is based on the standard error of the difference between two means,can only be used to test differences between two means

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With more than two means, could compare each mean with each other mean using t-tests

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Conducting multiple t-tests can lead to severe inflation of the Type I error rate (falsepositives) and is NOT RECOMMENDED

ANOVA

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ANOVA is used to test for differences among several means without increasing the TypeI error rate

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The ANOVA uses data from all groups to estimate standard errors, which can increasethe power of the analysis

Why Look at Variance When Interested in Means?

ANOVA

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Three groups tightly spread about their respective means, the variability within eachgroup is relatively small

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Easy to see that there is a difference between the means of the three groups

Why Look at Variance When Interested in Means? •

To distinguish between the groups, the variability between (or among) the groups mustbe greater than the variability of, or within, the groups

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If the within-groups variability is large compared with the between-groups variability, any difference between the groups is difficult to detect

ANOVA

difference between the groups is difficult to detect

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To determine whether or not the group means are significantly different, the variabilitybetween groups and the variability within groups are compared

One-Way ANOVA and Assumptions •

One-Way ANOVA^ -

When there is only one qualitative variable which denotes the groups and only onemeasurement variable (quantitative), a one-way ANOVA is carried out

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For a one-way ANOVA the observations are divided into

mutually exclusive

categories, giving the one-way classification

ANOVA

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ASSUMPTIONS^ -

Each of the populations is Normally distributed with the same variance(homogeneity of variance)

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The observations are sampled independently, the groups under consideration areindependent ANOVA is robust to moderate violations of its assumptions, meaning that the

probability values (

P-

values) computed in an ANOVA are sufficiently accurate even

if the assumptions are violated

Introduction of Notation •

Consider

groups, whose means we want to compare

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Let

be the sample size of group

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For the simulated verbal IQ and genotype data,

, representing the three possible

genotypes at the particular locus of interest. Each person in this data set, as well ashaving a genotype, also has a verbal IQ score

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Want to examine if the mean verbal IQ score is the same across the 3 genotype groups

ANOVA

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Want to examine if the mean verbal IQ score is the same across the 3 genotype groups^ -

Null hypothesis is that the mean verbal IQ is the same in the three genotype groups

Within-Groups Variance •

Remember assumption that the population variances of the three groups is the same

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Under this assumption, the three variances of the three groups all estimate this commonvalue-

True population variance =

-^

Within-groups variance = within-groups mean square = error mean square =

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For groups with equal sample size this is given by the average of the variances of thegroups

ANOVA

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For unequal sample sizes, the variances are weighted by their degrees of freedom

Between-Groups Variance •

If the null hypothesis is true, the three groups can be considered as random samplesfrom the same population(assumed equal variances, because the null hypothesis is true, then the populationmeans are equal)

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The three means are three observations from the same sampling distribution of themean

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The sampling distribution of the mean has variance

ANOVA

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The sampling distribution of the mean has variance

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This gives a second method of obtaining an estimate of the population variance

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The observed variance of the treatment means is an estimate of

and is given by

Between-Groups Variance •

For equal sample sizes, the between-groups variance is then given by:

ANOVA

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For unequal sample sizes, the between-groups variance is given by:

Testing the Null Hypothesis, F-test •

Another way of thinking about this ratio:

ANOVA

F Distribution and F-test •

The F distribution is the continuous distribution of the ratio of two estimates of variance

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The F distribution has two parameters: degrees of freedom numerator (top) and degreesof freedom denominator (bottom)

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The F

• test is used to test the hypothesis that two variances are equal

ANOVA

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The F

• test is used to test the hypothesis that two variances are equal

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The validity of the F-test is based on the requirement that the populations from which thevariances were taken are Normal

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In the ANOVA, a one-sided F-test is used, why not two-sided in this case?

ANOVA Table

I-1 = 3-1 = 2, since3 genotype groups,

AA, AG, GG

Slightly more complicated as thesample sizes are^ not all equal (

-^1

ANOVA

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For the simulated genotype, verbal IQ data:

AA, AG, GG

not all equal (

-^1

• 18-1+13 -1) = 43

Sum Sq / Df =

Mean Sq

One-sided P value,“statistically significant”at 0.05 level

Assumption Checking •

Homogeneity of variance = homoscedasticity-

The dependent variable (quantitative measurement) should have the samevariance in each category of the independent variable (qualitative variable)

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Needed since the denominator of the F-ratio is the within-group mean square,which is the average of the group variances

ANOVA

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ANOVA is robust for small to moderate departures from homogeneity of variance,especially with equal sample sizes for the groups

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Rule of thumb: the ratio of the largest to the smallest group variance should be 3:1or less, but be careful, the more unequal the sample sizes the smaller thedifferences in variances which are acceptable