ANOVA

ANOVA

​Want to compare MANY (i.e ≥ 2) groups/populations to see if there are differences between them...

​Need some way of determining if centres of populations/groups are either:

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⊳close (so no difference between groups) or

⊳far apart (so difference between groups)

Compare between-group variance to within-group variance using F-ratio:​

​The F-ratio is calculated using an ANOVA Table:

​where...

​A decision (about whether there is at least one group/population that is different to the others) is made by comparing the F-ratio to a critical/threshold value F_c .

​

F_c is found from statistical tables using ​

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k - 1 numerator degrees-of-freedom, and

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n - k denominator degrees-of-freedom​

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​Decision: if F-ratio > F_c then reject H₀ (i.e. at least one group/population different).

​EXAMPLE: Below is a partially completed ANOVA Table. For this table, answer the questions that follow.

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How many treatments/groups are there?

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What is the total sample size?

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What is the within-group d.f.?

Fill in the Blanks

What is the within-group sum-of-squares?

Fill in the Blanks

What is the between-group sum-of-squares?

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What is the between-group variance (i.e. between-group mean sum-of-squares)?

Fill in the Blanks

What is the F-ratio?

Multiple Choice

What are the hypotheses corresponding to this table?

• $$H_0:\ \mu_1=\mu_2=\mu_3$$    $$H_1:\ \text{at least two means are different}$$  

• $$H_0:\ F\ =\ 0$$  

  $$H_1:\ F\ \ne0$$  

• $$H_0:\ \mu_1+\mu_2+\mu_3\ =\ 0$$   $$H_1:\ \mu_1+\mu_2+\mu_3\ne0$$  

Fill in the Blanks

What is the critical/threshold F-value?

Multiple Choice

What is the decision regarding the hypotheses?

• Reject H₀ (i.e. there is a difference between the groups)

• Fail to reject H₀ (i.e. there is no difference between the groups)

​EXAMPLE:

Fill in the Blanks

What is the within-group sum-of-squares?

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What is the Total d.f.?

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What is the within-group d.f.?

Fill in the Blanks

What is the between-group d.f.?

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What is the between-group mean sum-of-squares?

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What is the F-ratio (to 3 d.p.)?

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What is the critical/threshold value for F ( to 2 d.p.)?

Multiple Choice

What is the decision about the hypotheses?

• Reject H₀ i.e. the groups are not different.

• Reject H₀ i.e. at least two groups are different.

• Fail to reject H₀ i.e. the groups are not different.

• Fail to reject H₀ i.e. at least two groups are different.

Multiple Choice

Does this confirm that none of the groups are different from one another?

• No, all the confidence intervals are different.

• Yes, all confidence intervals overlap.

• No, the confidence intervals have a clear order.

• Yes, all the confidence intervals exclude 0.

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What was the standard error used to calculate these confidence intervals (to 3 d.p.)?

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What is the critical/threshold t value used to calculate these confidence intervals (to 3 d.p.)?