# CRIM 320 Lecture Notes - Lecture 7: Multiple Comparisons Problem, Standard Deviation, Confidence Interval

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8 Dec 2017

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Crim 320 Module 7 – Analysis of Variance

- ANOVA = analysis of variance

- ANOVA best thought of as an extension of T-tests

- Like T-tests, ANOVA starts by comparing differences in means between groups

- What happens when your grouping variable has more than two categories

o > 2 groups

- focus on one-way ANOVA

The Logic of ANOVA

- within-group variability

o spread around the mean (variances)

- between-group variability

o differences in group means

- based on these two estimates of variability, you can draw conclusions about the

population means

differences in means – does that represent real group differences in the population

Procedure

- first question: are there differences in means across the groups?

o Usually there is some difference in means

- Chance: are observed differences attributable to random variability in the sample?

- OR

- True Difference: Is there reason to believe that some of the groups have different

values in the population?

Hypothesis Testing

- Null hypothesis the population means for all groups is the same

o Aka: there is no differences between group mean scores on the dependent

variable

- Alternative hypothesis: there are differences between groups

Mean

Mean

Mean

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- Note: The alternative hypothesis does not state which group differences are

significant

o Because there are more than two groups

One-Way ANOVA

- called one-way because it utilizes one grouping variable

- variable used to form groups is often referred to a factor

- what level of measurement will factors normally be?

o Grouping variable : creating GROUPS = nominal

- What level of measurement do you expect for the dependent variable?

o Interval : comparing means

Assumptions

1) DV is normally distributed for each of the populations as defined by the different

levels of the factor

a. Normal distribution for ALL groups

2) Variances of the DV are the same for all populations

a. Equal variances

b. Requires using Levens Test

3) Random sample

4) Independent of one another

How does ANOVA test for group differences?

- calculate the F ratio

- F = between-groups mean square / within-groups mean square

- Must calculate the between- and within-group variability (mean squares)

Estimating within-group variability I

1) compute within-groups sum of squares

a. obtain variance for each group (SD squared)

i. standard deviation squared

b. multiply each variance by (Group N -1)

c. sum up

2) Compute the degrees of freedom

a. Sum of (Group N-1) for all groups

i. There are 2 df

3) Finally, divide the sum of squares by its degrees of freedom

4) WMS – Within Mean Square

a. Average of the variances in each of the groups, adjusted for the number of

observations in each group

Estimating Between-group Variability I

1) compute between-groups sum of squares

a. subtract overall mean from each group mean

b. square each of these differences

c. multiply the square by number in each group

d. sum up

2) compute the degrees of freedom (df)

a. number of groups -1

3) divide the sum of squares by df (step1 / step2)

4) BMS – Between mean square

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