Statistics ANOVA https://www.123rf.com/photo_6622261_statistics-and-analysis-of-data-as-background.html

Difference Between Means We use the t-test when we have paired data because it is more powerful We could use it for other two-group comparisons, but we usually use another analysis:

Difference Between Means Comparing Several Group's Means: “ANOVA” “Analysis of Variance”

Difference Between Means t-tests can only be used for comparing two groups ANOVA can be used to compare two or more groups

Difference Between Means A paired t-test is more powerful A non-paired t-test is THE SAME as an ANOVA (ANOVA’s Excel output page is better)

Difference Between Means “ANOVA” stands for “ANalysis Of VAriance”

Difference Between Means The analysis assigns the variability in the data to: the difference between the groups the difference between individuals

Difference Between Means Sir Ronald Aylmer Fisher

Difference Between Means ANOVA sheet Salaries for Criminal Justice Jobs

Difference Between Means There are four classifications of jobs: probation, administration, correctional and patrol We want to compare the salaries to see if they are the same

What would be a good value for α? ANOVA PROJECT QUESTION What would be a good value for α?

α = .05 What would be a good level of practical significance? ANOVA PROJECT QUESTION α = .05 What would be a good level of practical significance?

ANOVA PROJECT QUESTION What is Ha?

μprobation ≠ μadministration ≠ μcorrectional ≠ μpatrol ANOVA PROJECT QUESTION Alternative hypothesis Ha: There are differences in the salaries of the four job classifications: μprobation ≠ μadministration ≠ μcorrectional ≠ μpatrol What is H0?

μprobation = μadministration = μcorrectional = μpatrol ANOVA PROJECT QUESTION Null (no difference) hypothesis H0: There is no difference in salaries for the four job classifications: μprobation = μadministration = μcorrectional = μpatrol

Difference Between Means Our strategy: We hope to disprove H0 and thereby to prove Ha

Why can’t you use a t-test for this data? ANOVA PROJECT QUESTION Why can’t you use a t-test for this data?

Difference Between Means use: Data Data Analysis ANOVA: Single Factor

Don’t include “Years”

Difference Between Means The first output table is just descriptive statistics:

Difference Between Means The second table is the ANOVA table … EEK!

Difference Between Means Don’t panic! Just look at the P-value

Difference Between Means The P-value is the likelihood of our pattern of differences in the means IF H0 was TRUE

Difference Between Means The probability is 1.99 E-5 or 0.0000199 (0.00199%) Is that very likely?

ANOVA PROJECT QUESTION We said we would reject H0 if it was only 0.05 (5%) likely to be true Can we reject H0?

Difference Between Means Remember – Reject the null hypothesis if the statistic is smaller than 0.05

Difference Between Means YES! If we are willing to be wrong in rejecting H0 5% of the time, 0.00199% is a whole lot less likely to be wrong!

Was the difference practically significant? ANOVA PROJECT QUESTION Was the difference practically significant?

What is your conclusion? ANOVA PROJECT QUESTION What is your conclusion?

Questions?

Difference Between Means For the CJ job classifications, we rejected H0 and concluded the salaries are different

Difference Between Means But… Are they all different, or is just one different or two or …

Difference Between Means Do a HI-Lo-Close Confidence Interval graph! http://markvanderpool.naiwe.com/files/2013/01/bigstock-Bright-Idea-Kid.jpg

Difference Between Means

Difference Between Means

Which means are different? ANOVA PROJECT QUESTION Which means are different?

Difference Between Means Now do you see why we’ve been doing Hi-Lo-Close graphs once a week since we learned them?

Difference Between Means BTW: pre-Excel, this comparison used to be REALLY hard to do!

Difference Between Means Yay Excel!

What would happen if you had a bigger sample size? ANOVA PROJECT QUESTION What would happen if you had a bigger sample size?

Difference Between Means PROJECT QUESTION What would happen if you had a bigger sample size? You would be able to show more statistically significant differences

ANOVA PROJECT QUESTION

Difference Between Means t-tests and ANOVAs are designed to be VERY powerful for small sample sizes

Difference Between Means That’s why we include a level of practical significance

Difference Between Means Similar to previous tests, the P comes from a standardized F-distribution:

Difference Between Means Because “z” and “t” are based on 𝒙 , they have similar shapes F is based on a variance, so it is in squared units!

Questions?

ANOVA The ANOVA we just did is called a “One Factor ANOVA”

ANOVA The ANOVA we just did is called a “One Factor ANOVA” Because there is only one category (type of job) – called a “factor”

ANOVA You can have as many factors as you want

ANOVA Excel can handle 2 factors

2 Factor ANOVA This one is tricky – you have to have the same number of observations in each category Educ Level HS Assoc Bachelor's M $ 15,000 $ 25,000 $ 35,000 $ 14,000 $ 24,000 $ 34,000 F $ 12,000 $ 32,000 $ 43,000

2 Factor ANOVA With only one observation it’s called “without replication” Educ Level HS Assoc Bachelor's M $ 15,000 $ 25,000 $ 35,000 $ 14,000 $ 24,000 $ 34,000 F $ 12,000 $ 32,000 $ 43,000

2 Factor ANOVA With only one observation it’s called “without replication” With more than one, it’s called “with replication” Educ Level HS Assoc Bachelor's M $ 15,000 $ 25,000 $ 35,000 $ 14,000 $ 24,000 $ 34,000 F $ 12,000 $ 32,000 $ 43,000

2 Factor ANOVA An ANOVA table: ANOVA Source of Variation SS df MS F p Total 1,214,916,667 11   Gender 52,083,333 1 7.35 4% Educ Level 960,166,667 2 480,083,333 67.78 0% Interaction 160,166,667 80,083,333 11.31 1% Within 42,500,000 6 7,083,333

Questions?